The planes are a fantastic place that offers lots of possibilities for interesting locations an adventures. But I guess that the philosophies on the planes should have an important impact, even (as in the case of my campaign) when the story does not center around them.
So how can I (playing in post-FW) create the feeling for my players, that the right belief is at least as important as a better new magic weapon?
The right belief can open doors for you what a sword never will.
On the planes, shared beliefs can create bonds as powerful as (or often more powerful than) blood.
The correct philosophy (or enough of an understanding that you can fake it) may very well save your life - the correct philosophy may save your soul.
Look at the Astral Plane - can there be any better proof that belief is power, than the boddies of dead gods?
If belief can power gods, imagine what sort of things people are attempting to believe into existance at any given time. Imagine what happens when those things come true.
The balance of the planes themselves rely on a balance of belief. If any plane should become dominate, the entirety of creation will feel the impact.
Everything a person does is at the mercy of their beliefs. Change a person's beliefs and you change who he is. (Sometimes quite literally)
Killing a man won't change his beliefs, nor will it reduce the power of those beliefs - in fact, sending a person to his plane of alignment as a petitioner may only serve to strengthen that plane; thus death may serve your enemies more than yourself.
A person who understands and believes may discover new ways by which to interact with the universe itself - he may find new ways to influence it as well.
People will follow a belief long after their armies are crushed and the borders of their worlds redefined.
I hope some of this helps.
Just because the factions have been ousted doesn't mean the power of belief has waned. (though, I'm not quite certain what the attraction of a post FW game is ... could you enlighten me?)
The point of the matter is, the answer to your question doesn't change post-FW. It only becomes more open-ended. You'll be far more likely to see esoteric sects and beliefs being represented in Sigil than ever before, and as Clueless pointed out...the Factions are still there, in one way or another.
Mechanics-wise, I like to use the belief point system from the Planewalker's Handbook, which gives you a sort of hand-on quick & dirty system to tackle the everyday effects of belief in the PCs lives. Basically, you hand out 1-3 points (1 for a minor sacrifice for upholding your beliefs, 3 for an ultimate sacrifice - rarely, if ever, awarded) per session (or more rarely, to make them special), which the PCs can use as automatic success in a single die roll. This goes a long way to rehabilitate the "+5, vorpal" crowd towards Planescape thinking, in my experience.
But mostly, I keep it in the flavour. PCs with strong beliefs get to see the effects of their beliefs reflected in the campaign world, plot hooks and NPCs revolve more around them, and big things and events are drawn to them.
Folks don't stop believing in their faction belief just because they move out of Sigil. That must be the top 'logical disconnect' I see in interpretations of post FW Planescape.
Want Hardheads? Go to Arcadia.
Want Guvners? Go to Mechanus
Want Doomies? Go to the Negative Quasielemental planes
Want Indeps? Go to the Outlands
Want Sensates? Talk to the Entertainers Guild in Sigil
Want ex-Mercykillers? Talk to the Sons of Mercy and the Sodkillers
Want Dusties? Go to the Mortuary, they still run it
Want Bleakers? They're still at the Gatehouse
All these factioneers didn't just *vanish* post FW. They didn't stop believing. They just stopped holding office!
Now they work in the background: they lobby, they fund guilds, they perform services - they just don't make laws or manage the city itself or push too hard to cause conflict. Saying they're 'gone' is something akin to saying that 'separation of church and state' means that there is no Church. Their attention has moved to the Planes as a whole. It's just become a much bigger playing field.
Ahh! I thought with the factions being banned from Sigil they were also forbidden to continue spreading their belief in the cage. Just as if e.g. the mojority of Cagers now would become Fated, they could try again to remove the Lady. :shock:
OK... so now they are like the sects, or something?
(Thinking about it, my initial assumption was not very logical. 
)
I'm confused; losing the political power in a city is no great loss, if you can still draw its citizens and use it to spread your ideas. And that's what the factions want, isn't it?
Well, drawing attention towards philosophical concepts should not be too hard then (hopefully).
But my major problem is to let the belief work. The idea with the reward points is quite okay and probably works, but its mayhap too mechanical. I don't know... the first impression I got from Planescape was 'surreal in optic appearance' closely followed by 'mature and philosophical'. While the first is relatively easy (even more with the great help I got in my art-thread ), the philosophical aspect seems to lose alot of its fascination, when it depends on game-mechanic to get my players interested in it at all.
And how can someone do something cool and impressing? Just to show some Belief Spell-like abilities (if such a thing exists) or Faction PrC special ability working seems a bit weak.
Perhaps I should let a likeable NPC talk to their characters and let her/him explain, why a blood needs a belief. This hopefully leads them to decide, what their characters stand for and makes them sensible for the ideals of other groups and gives a possibility to show them the faction's beliefs.
It's a great loss to those who were after political power in the city. That's why the factols got mazed - political power was more important to them than belief. Apparently.
The classic example of the power of belief in action is having the PCs trapped in a Gate-Town on the Outlands that leads to one of the Lower planes - they're trapped because the Gate-Town is sliding in to the Lower Plane. The balance of those interested in benevolent ends and those in malevolent ends has tipped in favour of evil. It's up to the PCs to do as many good deeds as they can and convince the locals to do likewise in order to correct the balance by winning over the beliefs of the people in that town.
Another idea is for the PCs to encounter a massive army of strange and alien creatures on the Outlands. The creatures use magical or psionic means to communicate with the PCs and ask them to join them in council. If the PCs do not accept, the creatures will chase them away with spells and blades. If they do accept they find themselves escorted to an artificial bowl-shaped valley that is swarming with these creatures.
They have been living on the Outlands since before the Guvners began keeping track of time and this is the second occasion that a roving tribe has stumbled on to the rings of the Outlands within sight of the Spire. The race is in the process of incubating a new Queen to give them guidance on how to survive and thrive in this new environment. That is why they are stopping those passing by to acquire knowledge and guidance.
In the centre of the bowl is a large egg and creatures from all manner of races and Faction outlook. The creatures inform the PCs that they will be given a turn to share information with the gestating Queen. Their first speech will be favourably accepted (if they don't mess up to badly). But as time goes on they realise that the race is intent on conquest and domination - at least that is the path that some fiendish councilors are leading it down.
It is now up to the PCs to convince the Queen and her tribe to follow another path. If they fail the creatures will go on to do untold damage in a year-long pilgrimage of conquest and enslavement. If the PCs convince the Queen to do otherwise they may win an ally for the forces that support the PCs.
That idea got a bit long-winded. But hopefully you can see that the power of words and action can sway events one way or another and have far reaching consequences. Structure the conflicts and moral dilemmas that the PCs encounter around a clash of beliefs and you soon have a very Planescapy-style campaign.
Unfortunately, when it comes to playing the game - there's no way around it. If you want to let belief do more than what belief can do in the real world (which admittedly is a lot already, and with the right plot lines you don't have to even touch belief-powered abilities), then you've got to have some mechanic to m onitor the power. Players have to have some way of letting their actions directly influence events - otherwise, without knowing how to go about it one of two things happens:
1) Without knowing how to trigger a belief power - they never do - and you're back at square one with players effectively ignoring it unless they get some personal satisfaction out of game from discussing and roleplaying it.
2) Without knowing if they can trigger a belief power - they constantly ask "Can I do X now? Huh? Can I? Can I?" Which gets annoying as heck.
You've got to give them a mechanic, but then everything has a mechanic. Spells don't lose their flash, or feats their nifty b/c the players area aware of the function behind them do they? Most players can put such metagame things to the side. Being able to do extra things with the belief also functions as a cookie - a reward that the players can feel substantially.
It's really not as weak as you'd think. It plays directly on human nature. It's aimed at the player, not the characters. Think of it as the carrot out of game. You'd be surprised how often just letting someone see something 'nifty' that they have no idea (for the moment) how it happened will prompt them to ask how they can do it themselves. Combined with a cool character doing it as you mentioned below, then the players start chomping at the bit - because you've intrigued the *players* not the characters. 
Treat it casually, make it intriguing, just don't make it obvious that it's a carrot and natural curiosity should take over from there.
What you're trying to do is get out-of-game motivation for your players to take the first step. After that, they should be having enough fun with the added depth that they do it one their own and you shouldn't have to offer as many carrots or cookies as rewards.
It's one way to do it. My game experience indicates that explosure to a really cool NPC tends to react that way - a brush with Rhys inspired a Cipher in one of my games for example.
You may need to make a judgement call on your players though - at this point you're trying to interest players, not just characters, so you gotta know what's best going to make your *friends* get interested. Without the players themselves being interested, the characters will only be doing it half-way anyway. In my group - it's nifty things they can do that gives a gut level 'I wanna try this out!' - we're also a very curious bunch. Your's may vary, so you may need to use a different carrot to get them going, and different cookies to reward them with. You're probably in the best position to figure out what will work.
Except Rhys. Because she's awesome.
I feel I have a clueless view of the things. Is there a significant difference between, let's say, joining the Harpers to defend the Dalelands from Drow and to join the Harmonium to enforce peace through uniformity - asides from the obvious differences between the settings??
The Harpers are a loose community of individuals pursuing a common goal on one Prime world (from what I know). The Harmonium united their world under one set of beliefs and then spread outwards to on to the Planes and other worlds.
Whilst the Haprers are happy for people to live out their lives as long as they do not threaten others with conquest, tyrrany or other evils. The Harmonium want to actively dictate how each and every person should live. Not only that but they also want to reshape the Outer Planes so that everything conforms with their beliefs.
On the Outper Planes belief is power. Some have accussed the Harmonium of causing one of the layers of Arcadia to slide over in to Mechanus (due to an abudance of belief shfting from Law with Goodness LNG to simply Law LN). But others dispute whether a relatively new Faction could achieve such a thing in a short period of time.
Regardless of the situation with Arcadia - the Factions and others know that by influencing the beliefs of those on the Planes and Prime worlds that they can bring about major changes. Even for larger organisations than the Harpers, the thing that sets Factions apart is their aim to reshape the Multiverse through belief. Some Factions exist almost exclusively to counter other belief systems but each believes them to be the correct view of the Multiverse.
So on the surface a groups of characters working for the Harpers and Harmonium may be carrying out similar tasks. If you take a step back and look at the bigger picture you realise just how very different they are.
I've found it! 
The perfect instrument of introducing philosophical aspects in my game and (hopefully) to make this kind of stuff interesting:
Deis, as my bashers have named it, a Rogue Modron that came from somewhere on Mechanus (it does not even really know itself where it originates), bringing *lots* of curiosity and many questions with itself the cutters have to think about themselves now in order to answer.
I know, rogue modron is not really very original, and it was created spontaneously by me because I was going to run out of preparation, but nevertheless my friends like it quite alot.
By creating its stats I realized thatit technically already knows more about the planes than the rest of the group together... 
But that is going to represent Deis' ability to learn very fast. 
That everybody has to have a belief should mean, almost every NPC is somehow associated with a faction, at least by sharing their worldview, aye?
The first step is deciding that's what you want to do.
The second step is convincing *them* that's what they want to do.
And then keeping your trap shut when they start talking about it in game.
But, for actual *techniques* to encourage this approach. Hm.
The easiest and standard way is to use the factions (no they're not gone after FW, just moved location) but that's more of a setting excuse for it. The biggest flaw with just tossing the factions out without any GM guidance, is that if your players aren't feeling it they may treat the factions as "just another oddball group" and not get that this what *they* are supposed to do. There's a simular vulnerability with clerics and churches too actually.
One way I've seen to get people actually *interested* in the belief mechanics is to let the players see someone do something that's "incredibly cool". And when they ask, they find out the only way to *do* that is via belief... that gets them interested in it.
Once they *are* interested though, you need to find a good way to 'measure' their beliefs. A player can (and should) have their own interpretations of the belief and how it should manifest in their actions. This may not be *obvious* at the table though since that sort of thing is very much inside their head. You've got to give them the chance out of character, and preferably in private, to explain their motivation. Once they start considering their motivations, that should help them out considerably - and gives you a chance to reward them for it and gear the story to play off of it more.
You need to give your players the feeling that their own philosophies really have impact on what's around them. You may want to toss them into a 'grey' situation - where their words and suggestions influence the decisions of others and there *is* no clear answer. Just make sure that the people they're talking to put up enough resistance in the debate that your players really get into the depths of it, without putting up so much resistance that the players give up.
Alternatively, if you have at least one player who already does this sort of thing - talk to them in private and conspire to spread that philosophy around.
Then, give them opportunities to shine. For example:
In a recent game I played in, my GM had us running to get stop an ally of ours from committing suicide after her whole life fell apart. She was sitting on the edge of a diefic domain, and when we teleported in - we were shunted into the domain itself to explain to WeeJas (or her proxy) exactly what we were doing. Then, given her status as a power of law and death and magic, we had to explain why we *should* be allowed to go 'save' our friend from her own choice, justifying our intervention and our actions since it's not a given that one actually has the *right* to interfere in anothers death - at least not to Wee Jas.
That done, once we found our friend, it was then *another* conversation to convince her to stay alive - as this close to the domain, we couldn't just clonk her over the head and take her someplace more sensible to talk.
There's nothing quite as fun as seeing a Doomie of the 'break-things' kind, trying to talk someone else *out* of suicide. In her case it came down to: "Everything falls apart. Including your enemies. There's no need to do this *now*, don't you want to live to see them down and out too?"
Our local psuedo-existentialist/Bleaker came out with: "Right now there is nothing, absolutely nothing more that they can do to you. They've taken all meaning and purpose from you - but now, with nothing left - you have everything to gain. You can't go any further down than this. And now - it's truly *your* choice what meaning you want to make in your life. You're stronger than you've ever been."
That's the sort of scenario and arguement that gives the best opportunity for your players to explore what they're doing.