I was wondering exactly this this morning! There are a few hints in the orc conversations that they are immortal. Gorbag and Snagrat talk about the "good old days" as if they were a long time ago and they also seem to remember the "Great Seige" (ie, the Last Alliance) as if they were there.
The most interesting part is that they regard the Nazgul as newcomers! ("They're His favourites nowadays": how long is it since the Nazgul were not Sauron's favourite servants?) That would make them very, very old indeed.
There's also old Grishnahk to consider: the level of trust put in him by Sauron seems extraordinary; more than the Mouth of Sauron or any other mortal. Surely only very long service indeed could bring an orc up to the point where Sauron actually trusted him with the Ring? How long I don't know but given the nature of Sauron and orcs I'd have guessed centuries at least.
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