Harry's learning of the Patronus (or rather, his initial struggle to do so) is sad in a way that's so obvious that it's almost subtle. Especially the film, where Harry fails the first time and Lupin asks him what happy memory he chose to try to power the Patronus. Harry responds with his first memory of riding a broomstick and Lupin says that it wasn't "nearly strong enough." But until Harry finds the loophole (the memory doesn't always have to be real — it can be a positive hope or dream or the like), that's literally the best he can come up with up to that point.
Not necessarily the best he has (he could probably think of his friends, and it would strong enough in time), but what's sad is that the happiest thing he can come up with is something that he can't even really remember, can't tell if it's even real. That Harry's happiest memory is his parents, that it is just of them doing nothing but talking... now that is the Tear Jerker. How often he must have yearned for them, like any orphan (or even anyone, when their own parents die)...
A rather subtle, but effective, moment comes in Prisoner of Azkaban when Harry catches himself half-hoping to be overwhelmed by a Dementor since, horrible as the experience is, it's the only time he's ever heard his parent's voices.
Buckbeak's execution by the Ministry of Magic especially with Hagrid's friends nearby and powerless to help. Thank heaven — and a Time-Turner — that they had more power to stop it than anyone realized.
Even worse is how helpless Hagrid is.
Harry's Hope Spot when Sirius tells Harry he's his godfather and offers Harry the chance to move in with him, just to have it snatched away. For a brief moment it seemed his dreams were coming true and he could leave the Dursleys, just for reality to come back. No wonder he was so despondent despite saving Sirius and Buckbeak's life.
Hagrid losing his confidence because of the incident with Buckbeak. Even worse is how Malfoy did it on purpose!
Lupin leaving, especially when you consider just how many jobs he must have lost in this same way. No matter how good he is at the job, or how well liked he is by his co-workers or pupils, it always comes down to how long he can keep his condition a secret. All it takes is one person letting slip the fact that he's a werewolf, and he's fired/has to quit because of Fantastic Racism.
In the film, Gary Oldman's portrayal of Sirius seems, for the most part, rather Narmy. However, when he screams "I DID MY WAITING! TWELVE YEARS OF IT! IN AZKABAN!", it becomes clear how broken he is.
Hermione's (relatively minor and thankfully averted at the end of the novel) Sanity Slippage because of how much work she's taken on. She loves school and she loves learning and she wants to remain top of every class, but the poor girl is clearly cracking under all the pressure. Add in her spats with Harry and Ron, her trying to help Hagrid defend Buckbeak, and the fear of a mass-murderer being on the loose and after one of her best friends, and it's frankly a miracle she's still sane by the end of the year!