There is no clear answer, because the origins of Orcs are murky. Assuming, that, corresponding with the published explanation of the origin of Orcs, they came from Elves, I propose two possibilities:
They may have in origin been Elves, but Morgoth quickly interbred them with Men when they awoke and began worshipping him. All the immortal Orcs were killed in the wars of the First Age and only Orcs mixed with Men survive. These technical half-Elves may get long life, but not immortality. The immortal Orcs never come back because if Feanor is too evil to be reincarnated before the End, then the average Orc most definitely is. That said, some of the evil spirits at Sauron and Morgoth's command are, perhaps, the spirits of killed immortal (or mortal) Orcs.
But as an alternative, I would say that from Elves or not, their fates may have been changed by Eru and Morgoth. Consider what Finrod and Andreth claimed about Men:
'Yet among my people, from Wise unto Wise out of the darkness, comes the voice saying that Men are not now as they were, nor as their true nature was in their beginning. And clearer still is this said by the Wise of the People of Marach, who have preserved in memory a name for Him that ye call Eru, though in my folk He was almost forgotten. So I learn from Adanel. They say plainly that Men are not by nature short-lived, but have become so through the malice of the Lord of the Darkness whom they do not name.
'That I can well believe,' said Finrod: 'that your bodies suffer in some measure the malice of Melkor. For you live in Arda Marred, as do we, and all the matter of Arda was tainted by him, before ye or we came forth and drew our hroar and their sustenance therefrom: all save only maybe Aman before he came there.1 For know, it is not otherwise with the Quendi2 themselves: their health and stature is diminished. Already those of us who dwell in Middle-earth, and even we who have returned to it, find that the change3 of their bodies is swifter than in the beginning. And that, I judge, must forebode that they will prove less strong to last than they were designed to be, though this may not be clearly revealed for many long years.
'And likewise with the hroar of Men, they are weaker than they should be. Thus it comes to pass that here in the West, to which of old his power scarcely extended, they have more health, as you say.'
(HoME X)
So until the Fall, Men lived longer and and in greater health. The 'malice of Melkor' affects even the Elven bodies, which don't last as long as they should - thousands of years instead of to the End.
Andreth even believed that Men were not supposed to 'die' at all, but rather ascend to heaven bodily.
So I would propose this: the 'malice of Melkor' quite naturally afflicts the Orcs most severely. Even if they were intended to be immortal (by their potentially Elven nature) their bodies simply fall apart. So they look mortal to us, and we have no way of determining whether their spirits linger in Middle-Earth (like the Elves) or go on beyond the world (like Men.)
It must be added, however, that some Orcs were immortal:
This last point was not well understood in the Elder Days. For Morgoth had many servants, the oldest and most potent of whom were immortal, belonging indeed in their beginning to the Maiar; and these evil spirits like their Master could take on visible forms. Those whose business it was to direct the Orcs often took Orkish shapes, though they were greater and more terrible. Thus it was that the histories speak of Great Orcs or Orc-captains who were not slain, and who reappeared in battle through years far longer than the span of the lives of Men.
