Geek here.
The Elves have spirits (although perhaps not souls as we would define them) that proceed to the Halls Of Mandos upon death. The Halls are within the circle of Ea (Earth) and therefore are not precisely analogous to heaven. Elvish spirits may be rebodied by the will of the Valar, but with one exception (Luthien) may not return to Middle Earth. Elves are bound to Ea, and likely even thier spirits will perish with it were it to perish, although this is not explictly stated in the texts. Based on the statements of Galadriel, the Ents may share a common fate with the Elves.
Human souls proceed through the Halls Of Mandos upon death, not stopping until they have breached the Door of Night and attained the void where Eru dwells outside the circles of the world. From thence, no one knows what becomes of them, and none have yet returned who passed the Door.
It is not written that Dwarves or Orcs have any immortal parts. The fact that they are lesser orders of creation, having been created by the works of the Valar rather than Eru, seems to reinforce this notion.
Geek out.
Glorfindel was an exception as well. Tolkien wrote that the Glorfindel who died killing a balrog in the fall of the elves' hidden city (the name of which escapes me at the moment) was the same Glorfindel as met Aragorn and the hobbits at the Ford of Rivendell.