(Disclaimer: the following will have various spoilers for Harry Potter and the Prisoner of Azkaban book and movie, as well as bits of the Christopher Nolan 'Dark Knight' movies)
First and foremost, it's kind of a pet peeve of mine when people throw around the phrase 'plot hole' in ways that aren't actually a 'plot hole', as if to say, 'Yeah, I don't actually know what this means but it sounds cool'. There are varying definitions to this term, but at the end of the day, in order to be a 'plot hole', said 'hole' must actually go against the plot itself.
A lack of information, as long as that lack doesn't keep something from being able to happen, is not a plot hole. A great example on this is in The Dark Knight Rises; tons of people have called it a plot hole that Bruce Wayne was able to get back to Gothem after his time in the prison 'hole' (lol). However, simply not telling the audience how isn't a hole, it's just a lack of information, but that lack of information does not invalidate it.
It also helps that this is a series that has already setup that Bruce Wayne is a man of considerable resources and talents. Batman Begins already showed us a man who gave up his life and wealth to travel from country to country without a cent to his name, or his name itself (speaking up about who he was could have saved him some trouble when 'robbing' a warehouse of Wayne Enterprises goods).
So as a major fan of the Harry Potter series, both in movies and books, it makes me scratch my head severely when people call time travel, and the Time Turner itself, a massive plot hole in the Harry Potter universe. The main argument is that there shouldn't be any problems, any deaths or loss of any kind, in a world that has a Time Turner. But in my opinion, this can only be said by someone who just hasn't paid close enough attention to what was shown, in book AND movie form (though I will admit that the former did a better job).
The principle of the Time Turner is that it is a device that can take the wearer (and anybody within it's zone, for lack of a better word/term) back in time a short distance; one hour per tern, with no mention of a maximum that I can remember. However, the idea of the Time Turner is NOT about 'changing' the past, it's actually about fulfilling what was already done.
When Harry and Hermione go back in time during the 'climax' of the story, they do not do anything that wasn't already done in the first place. Some may have thought, at least at first, that poor Buckbeak was truly killed that first time going through the events, but he was not. After all, Harry and Sirius were not sucked dry of their souls the first time through because future Harry was already there to cast the Patronus that saved them.
Had 'future' Harry not been there that first time, there would have been no Harry left to go back in time in the first place.
The Dumbledore that approaches Harry and Hermione in the hospital wing of Hogwarts was not a Dumbledore that had witnessed the beheading of Buckbeak, but a Dumbledore that had already witnessed a mysterious disappearance of Buckbeak. This Dumbledore had also heard tale of a mystery person who summoned a powerful Patronus that chased off a pack of Dementors, and these two mysteries combined, in my opinion, to tell Dumbledore that Harry and Hermione MUST have gone back in time and that they were responsible for these events (Ron was suffering a broken leg, in the book at least, so he clearly couldn't have been involved).
In regard to those who have asked why Dumbledore would have been so cryptic with how he worded to Harry and Hermione what they had to do, I can only tell you what I have always thought. In my opinion, Dumbledore couldn't just come out and tell them what they already did, because they hadn't done it yet. If Harry and Hermione had gone back in time with the knowledge of what they were about to do, they could have very easily gotten cocky and reckless with their actions, thinking that they couldn't possibly screw up, since they had already done it.
This recklessness could have caused them to be careless, seen or otherwise mess up.
Another example which is only in the book has Hermione missing a class. For those who only saw the movies, a very basic overview is this; an exhausted Hermione falls asleep at a desk as she was trying to study, and is awoken by Harry and Ron who tell her she missed a Charms class. She is mortified over her carelessness and goes on a tangent of how she hopes Flitwick will understand and let her make up her work.
I over simplified this, but that is the basic idea of what happened at one point in the book. She doesn't just rush off to the bathroom or something to go back in time, because she already missed that class. Her absence has already happened, and she cannot go back and change the past. Time Travel in this universe is not about changing the past, but instead it's about fulfilling what already was.
Finally, at one point in both the book and the movie, Hermione gives a very brief explanation of the rules of time travel to Harry, once they had already gone back in time. She tells Harry that 'bad things happen' to people who try and meddle in time. There is no real information provided as to how or why this is, the character herself probably wasn't even told, but just remember, as said earlier, a lack of information in of itself is NOT a 'plot hole'.
In closing, I would just like to say that ultimately, J.K. Rowling is the author of this fantastic world, and the soul authority on what she intended. So if she were to ever read this, and tell me I got it wrong, I would have no choice but to accept that. However, as it stands right now, this is how I see the concept of Time Travel in the Harry Potter universe.
Oh, and did I mention, I absolutely LOVE the Harry Potter universe?