A generally accepted rule when it comes to Time, is that in no way should one mess too bad with the timeline. A fact, that was indeed considered to be true even in the Harry Potter universe.
But, then we see Harry and Hermione using the Time Turner in The Prisoner of Azkaban to save Sirius and Buckbeak from certain death.
How was this possible or admissible? Shouldn't they be able to bring back, say Harry's parents or even go back in time and kill Voldemort when he was not a threat?
Hermione explains both that they are breaking the Time Turner rules and some of the principles behind those rules.
“Hermione,” said Harry suddenly, “what if we - we just run in there and grab Pettigrew-”
“No!” said Hermione in a terrified whisper. “Don’t you understand? We’re breaking one of the most important wizarding laws! Nobody’s supposed to change time, nobody! You heard Dumbledore, if we’re seen-”
“We’d only be seen by ourselves and Hagrid!”
“Harry, what do you think you’d do if you saw yourself bursting into Hagrid’s house?” said Hermione.
“I’d - I’d think I’d gone mad,” said Harry, “or I’d think there was some Dark Magic going on-”
“Exactly! You wouldn’t understand, you might even attack yourself! Don’t you see? Professor McGonagall told me what awful things have happened when wizards have meddled with time. ... Loads of them ended up killing their past or future selves by mistake!”
(Prisoner of Azkaban, Chapter 21, Hermione's Secret).
So they were breaking both the letter and the principle of the law in rescuing Buckbeak. They broke the letter insofar as the Time-Turner was only meant to be used for Hermione to get to her lessons, and it was on that basis that the Ministry allowed her to have it.
“It’s called a Time-Turner,” Hermione whispered, “and I got it from Professor McGonagall on our first day back. I’ve been using it all year to get to all my lessons. Professor McGonagall made me swear I wouldn’t tell anyone. She had to write all sorts of letters to the Ministry of Magic so I could have one. She had to tell them that I was a model student, and that I’d never, ever use it for anything except my studies..."
(Prisoner of Azkaban, Chapter 21, Hermione's Secret).
They were also breaking the principle of the Time-Turner by changing the past to reshape the future. As the question implies, this is generally regarded to be bad form (or, at least, risky) in most sci-fi. Harry Potter's time travel consequences are less extreme than other works of fiction. Buckbeak's personal timeline can be altered without the universe collapsing. All Harry and Hermione have to do is ensure that they don't run into their past selves.
Dumbledore knew that they were technically abusing the Time-Turner when he gave it to them. He judged that it was worth bending the rules in this instance since it was the only way to save Sirius's (and Buckbeak's) life. He emphasised only the importance of not being seen when changing the past. This was the only "law" he cared about observing, not the Ministry law not to change time in this fashion.
"Miss Granger, you know the law — you know what is at stake...You - must - not - be - seen.”
(Prisoner of Azkaban, Chapter 21, Hermione's Secret).
As for the point about using Time-Turners to rewrite distant history, this isn't possible. Time-Turners can only be used to go back a maximum of five hours into the past.
"As our investigations currently stand, the longest period that may be relived without the possibility of serious harm to the traveller or to time itself is around five hours...All attempts to travel back further than a few hours have resulted in catastrophic harm to the witch or wizard involved."
(Pottermore, "Time-Turner").
Time-travel becomes too dangerous and unstable when going back more than a few hours and so it's not allowed.
The case of saving Sirius is fundamentally different from going back in time to save dead people. Simply put, Sirius was fully alive and well at the moment that Harry and Hermione initiated the time travel. They were not trying to change the past; they were trying to change the future. The issue they were dealing with was not that Sirius had been wrongfully killed, but that he was about to be wrongfully killed. (Well, technically have his soul sucked out, but for the purposes of this answer we can simplify things.)
An important principle of time travel in the Harry Potter books is that whatever the characters do while back in time already occurred the first time around. This is most obvious – and indeed the point where Harry understands – in the case of the patronus. In the original timeline Harry was saved from the dementors by someone across the lake casting a patronus. At the time he did not realize who it was, but when he was in the past he realized that it was himself, and so he cast the patronus as his future self. That was not a change of the past, but a fulfillment of the past.
Given that this is how time travel works, the past cannot really be changed. If a character's future self would go back in time to save someone, that person would never have died in the first place. When someone dies in the original timeline that is a proof that no future characters will go back in time to save them.
We can now try to understand what the purpose of going back in time in Prisoner of Azkaban was. After all, if the past cannot be changed why would someone ever go back in time? The answer to that is, as we started with, that they were not planning on changing the past; they were planning on preventing the future. Nothing that had already occurred stood in contradiction to Sirius escaping. The only problem that they were facing was that in their current situation they did not have any means to save Sirius. Going back in time was not meant to change the past; it was meant to give them the ability to do what they would be unable to do as their present selves. Knowing the events of the three hour period enabled Harry and Hermione to position themselves so that they would be able to undetectedly help Sirius.
If you look carefully at Dumbledore's words, this becomes even more clear:
"What we need," said Dumbledore slowly, and his light blue eyes moved form Harry to Hermione, "is more time."
Note the second to last word. Dumbledore did not say that they need to change the past, or do any number of strange things to time. All he said was that they need more time. Sirius was about to have his soul sucked out, and there was no time for them to do anything to stop it. Going back in time was simply the means of gaining extra time to conduct the escape plan.
That all covers Sirius. With Buckbeak it is a bit of a different story. The events involving Buckbeak were certainly in the past at that point. Whatever had happened at his execution had already occurred three hours prior to the initiation of the time travel. As such, it should have been impossible for them to change anything. And indeed it was impossible. They did not change anything; they merely caused the fulfillment of events as they had initially transpired with the influence of future characters. That is to say that Buckbeak had never been killed, because Future Harry and Future Hermione had saved him.
However, as mentioned above, given that the past cannot be changed, what caused Buckbeak to be saved in the first place? There would be no motivation for Harry and Hermione to go back in time to save him, since he had already been saved. While it is true that Harry and Hermione did not actually know that Buckbeak had survived, Dumbledore was aware of this, so he shouldn't have had any motivation to send them back in time to save Buckbeak.
We can address this with what I alluded to in this answer. Bucbeak's escape was only incidental. If Dumbledore had not needed to help Sirius then he would not have sent Harry and Hermione back in time, and Buckbeak would have in fact been killed. Once they were going back in time Buckbeak became a necessary accessory for Sirius's escape, and therefore had to be saved. Because he had to be saved during their time traveling, he ended up being saved all along. When Dumbledore first saw that Buckbeak had escaped he was presumably confused. But once he realized that they needed to go back in time to save Sirius, it was obvious that the best way to save him would be by using Buckbeak. He thus figured out that it had been his plan all along that allowed Buckbeak to escape, and therefore hinted to Harry and Hermione to use Buckbeak in that manner.
So to conclude, no past events were changed in the act of saving Sirius. A future event was prevented, and the means of preventing the future event was what caused the past to happen the way it did the first time around. Time-Turners could not be used to change any other events in the past because that would exceed the scope of a Time-Turner.
Just to add to this answer, Dumbledore isn't confused by Buckbeak's escape initially, because he sees Harry helping Buckbeak get away out of the window. At this point (the first time around), he assumes that it is just regular Harry trying to help for Hagrid's sake, because he doesn't know that he will later send Harry back in time. Like Harry does when he mistakes his Time Turner self for his father, Dumbledore mistakes Time Turner Harry for regular Harry so no scary consequences of meeting other selves occur. Dumbledore wants Buckbeak to escape for Hagrid's sake and Harry not to get into trouble, so even without knowing about time travel or Sirius at that point, he keeps the executioner busy and buys the time they need. Later on, Dumbledore realises that the Harry he saw must have been due to time travel and sends Harry and Hermione back to complete the loop. By sending them back, he ensures that when future Harry and Hermione return to the present after the rescue, they don't end up with two versions of themselves in the same point on the timeline.
There are different types of time travel. In BTTF if you change the past, you create a new time line. This eliminates possible paradoxa. But in Harry Potter, the re-action causes the action which again causes the re-action, etc.
When Harry and Hermione thought that Buckbeak died, they used the time turner to rescue him. But Buckbeak never died. He was already rescued by Harry and Hermione from the future. But present Harry and Hermione didn't know about that, and this leads to the paradox. Sirius also never stayed trapped and was executed, due to the same paradox.
This also explains why after Sirius died, they couldn't use a time turner to rescue him, because you never actually can change the time line and if you stop Bellatrix from killing him, he would either die no matter what or the paradox would end the time line. But that's too complex to talk about now
