The biggest concern for the current fleet of U.S. reactors (mostly all Generation II designs) in terms of long operation is embrittlement of the reactor pressure vessel (RPV) due to radiation damage (mostly neutronic). Embrittlement of the RPV comes into play when severe accident responses (for either Design Basis Accidents (DBAs) or Beyond Design Basis Accidents (BDBAs)) dictate fast, extreme cooling of the RPV that can lead to pressurized thermal shock (PTS) events. The biggest hurdle toward getting approval is proving which-and-every way to a high confidence level that a PTS breach of the RPV will not occur from this embrittlement. If plants cannot do this, the NRC will not issue a license extension because the plant cannot prove its safety. If you care to read more on it, consult 10 CFR 50.61 for details (or the whole thing at the10 CFR 50 Part Index.
Are there other requirements? Yes (see the 10 CFR 50 index above). However, this is the one aspect I wanted to expound upon since turbomachinery has been replaced/upgrade, fuel is refreshed every 18 months or so, and piping is constantly checked. But I wanted to stress the safety issue. The NRC has 100% no qualms about telling a plant "no" if that plant cannot prove it is safe to operate.