[Obviously, the following strays away from canon (mostly in the form of Dungeon #100), which rather strongly suggests that githyanki souls tend to end up in the Lower Planes. The githyanki/zerai history/belief system never did fit very well with the Great Wheel cosmology, in any case. Reincarnation has also been discussed here.]
The Liberator's children do not spend much of their time or cultural energy on the contemplation of individual souls. Such matters are mainly in the realm of mystics, sages, and some scholars. A typical githwarrior spends much more time thinking about the war against the Great Enemy, and in many cases also the War of Two Skies, than about the mysteries of life after death.
When they do contemplate the afterlife, most githyanki believe that they are reborn (though perhaps "rehatched" is a more precise term) through a potentially endless series of reincarnations. The soul of a Githyanki, it is said, cannot rest until the very last of the enemies of the People is destroyed. This is commonly understood to mean all of the Illithid, but opinions differ on whether all the graith in the multiverse must be destroyed or conquered as well. The orthodox take "enemies of the People" to include all githzerai, but some heretics explicitly believe that the destruction of all Githzerai is unnecessary. Some of the most extreme zealots include githyanki of competing ideologies in the "enemies of the People" definition, as well. Some mystics claim that, in the (unlikely, and certainly not any time soon) event that all the ancient Enemies were to be destroyed at last, individual souls might be able to attain a state of peaceful rest, but githyanki literature on the subject does not describe a defined afterlife or plane in such a case.
Unlike some other peoples who believe in reincarnation, Githyanki do not believe in either karmic justice or karmic punishment: the slate is wiped clean at the moment of death. Those githyanki who are executed for various reasons are not punished in future incarnations, nor are heroes necessarily rewarded in future lives. They do believe that echoes of one life may be present in the next, such as a sense of deja vu, a particular mannerism, and so on. Githyanki do believe that souls are generally improved and made stronger over many incarnations.
The People also believe that it is possible for an individual to stop the cycle of reincarnation, but in all but one case, this is not to be desired. A githyanki who becomes apostate and worships a heathen god becomes, in githyanki eyes, the slave of that deity: his soul will travel to wherever that unclean god-thing desires. This is one reason (of several) why githyanki who worship foreign gods are so thoroughly despised; it is believed that they essentially desert the Great War by taking themselves out of the cycle of reincarnation. (In such cases, githyanki knights will often attempt to convince or force the apostate in question to recant prior to execution, rather than risk losing a soldier against the illithid in some future life.)
The githyanki belief in reincarnation is ancient, but since the ascension of the most recent Vlaakith to the throne, it has become a matter of orthodox belief that to become a recipient of the Queen's Honor (or the ritual of retirement, or the Great Reward, or whichever polite euphemism one cares to use) is the most elevated state -- or in the case of those with whom She is not pleased, the worst punishment -- possible. The soul is taken out of the cycle of reincarnations, but (when She is pleased) becoming a sword-spirit or similar created undead is a state somewhat akin to nirvana.
What exactly happens to the souls of individual knights, with their connection to the Lower Planes, might be different from those of most githyanki, but such speculations are regarded as one of the mysteries of the knightly orders. Many are not entirely sure about the souls of duthka'giths either. Perhaps Tiamat has an interest in them...
Githyanki generally believe that they can only reincarnate as githyanki. Some heretics believe that it is possible to reincarnate as githzerai (!!!), or perhaps pirates. Significant contact with Athasian gith is so recent that not much literature has been written on such a possibility. Even most heretics would not believe they could possibly come back as a human or some other disgusting barbarian species.
"Accepting the Queen's Honor, and creation of a sword spirit, do not hinder reincarnation. Why would the great queen take the souls of people from the Circle? Necromancers agree that the soul departs upon death and is not re-bound nor destroyed when an undead rises again. The simplest forms of mindless undead are animated by a portion of negative energy, similar but opposite to the life energy of a living being, but weak and without will. The more powerful, free-willed undead are animated by negative energy too, but in their case a template of the original soul and mind is created. Think of a shadow cast by a physical body. If it is a cloudy day, or the light is weak or defuse, there will be a shadow but it will be fuzzy and undefined. A clear, strong, steady light will cause the shadow cast to conform completely to the original form that casts it. Yet, it remains a shadow, and though a clear shadow may reveal much about the form that cast it, it cannot mimic it completely. So, though you may see fallen brothers fighting beside you, they are are both the same and very, very different from the children of Gith you once knew."
- Gar'kuul Suumn, leader of the Black Librum during the reign of Vlaakith CXXIX
***
"The queen take my body, but not my soul! That I can fight the Great Enemy for all eternity is my only desire."
"Selfish! That I serve the People is my desire, and if Her Majestsy drink my fountain dry in order for me to do so, then so be it!"
From The Redemption of Kwaan'sh'sh, by Sar'yel-Reth.