At the end of The Prisoner of Azkaban, Harry and Hermione travel back in time to save Sirius (and as it turns out, Buckbeak) and escape the Dementors.
If this is a legitimate option for saving people, why don't they ever use it again? If they can use it to save Sirius in The Prisoner of Azkaban, why can't they use it to save him when he dies in The Order of the Phoenix? For that matter, why can't they go back to the beginning and kill Tom Riddle before he becomes Voldemort?
The only thing I can think of is that in The Prisoner of Azkaban, maybe they use the Time Turner before Sirius is given to the Dementors, and so they're not actually saving him from something that already happened; they're altering a series of events that they know will lead up to him being handed over to the Dementors. Is that how time travel works in Harry Potter land? That is, you can't alter something as definitive as death (or a Dementor's kiss), but you can alter other events that have occurred and consequently prevent a death that you know will occur?