Depends on the barrel and cleaning between range trips. Quality of materials will also extend barrel life. The Army M24 barrel was said to have a 5,000 round barrel life, but we logged some past 10,000 rounds that were still less than 1 MOA. Some people also rely heavily upon barrel brake-in procedures and loss of life with out it (see other threads about that for personal opinion). If you get something below 300 rounds, its still fairly new...
I guess anything can wear out eventually. As far I know two things can definitely kill an action, repeated high pressure or a single incident setting back the actions locking lugs or the reciever cracking from firing or removing a barrel. As far as it wearing out from reasonable use I would say it would vary and be somewhere north of 25,000 rounds. Alot would depend on the quality of material in the actions construction.
Most "custom" actions (Defiance, Impact, Bighorn, etc etc) won't feel broken in until at least a couple thousand rounds. You'll shoot out a shit load of barrels before an action ever needs to be replaced. You'll need to replace the firing pin and firing pin spring and ejector/extractor at some point in its lifetime but other than that. As long as you keep up on regular maintenance, keep the action clean and rust free (if its not coated), I'd guess tens of thousands of rounds (30-50k) if not more.
I think how the action was maintained is more important than round count. In my mind An action with 1000 rounds on it that has rust and galling on the lugs would be worse than an action that has 25,000 rounds on it that doesn't.
I would venture to guess that the only ones that really have to worry about action burnout, would be whatever LE agency or contracting company buys military surplus precision rifles. I'm not an armorer so I cant really base that on anything other than just thinking about that life cycle.