This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Harry Potter and the Prisoner of Azkaban
While Harry obviously has the hardest time of the Trio (in dealing with the fact that a mass murderer is out to get him and said murderer was his parents' best friend), Ron and Hermione don't have it great either. Ron is dealing with the fact that his pet (unaware of its actual identity) is supposedly dying, something he doesn't get much sympathy for. Hermione, meanwhile, has been taking more lessons than usual. She's no doubt struggling under all the pressure, and it's frankly a miracle she's still sane by the end of the year!
When Harry realizes what his Patronus is.
"Prongs rode again last night."
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)...
Yet another tear-jerky item regarding the Patronus comes at the climax of the story: Harry, after the time jump, sees himself and Sirius being consumed by the Dementors, and Harry is waiting for his father to come and save him. He realizes his dad isn't coming and casts the Patronus himself. It's a really powerful scene with that in mind.
A rather subtle, but effective, moment comes when Harry catches himself almost 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.
Mind you, this becomes Fridge Horror when you learn that Harry has a piece of Voldemort inside him and realize that it isn't Harry's memory of his father telling Lily to take their son and run, and neither is the one of Lily screaming...it's Voldemort's.
Another somewhat subtle one is when Harry is learning how to cast the Patronus Charm and can't focus on a happy memory hard enough, he's struggling to find a happier memory each time until finally he ends up thinking that if something simple but happy can't help him, nothing will.
During Harry's first tutoring session with the boggart!dementor, Harry hears his mother and father's voices for the first time in a decade. When Harry tells Lupin about it, he freezes and whispers "You heard James?" He must have missed his best friend so much.
To go with the above, early in the book, Lupin stops Harry from facing the boggart on the assumption that it will take the form of Lord Voldemort. He nonchalantly states that such a thing would cause quite a ruckus and would thus not be a good idea. However, Harry greatly resembles his father James, Lupin's best friend...whom Voldemort murdered. It's quite likely that Lupin jumped in as quickly as he did because he was also trying to prevent having to see what would essentially be a near-reenactment of the night James died.
Hagrid losing his confidence because of the incident with Buckbeak.
J. K. Rowling intended the Dementors to represent clinical depression, so the description of Hagrid in Azkaban becomes especially harrowing. He wasn't there for very long, but the presence of the Dementors made him relive his worst memories: The death of his father, the abandonment of his mother, being expelled from Hogwarts, etc. It wasn't until he left that he returned to his usual jovial self. His terror of the place partly informs why he doesn't just help Buckbeak escape into the wild.
"I can't ever go back to Azkaban..."
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 him 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.
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.
The fact that James and Lily were killed by Voldemort because one of James' best friends since his first year at Hogwarts decided to betray them to the Death Eaters. Then Pettigrew decided to fake his death and frame Sirius for it once it led to Voldemort's downfall and Sirius had to spend most of his life in prison for something he didn't do while Remus went on thinking all of his friends were dead. No wonder Sirius and Remus wanted to kill him.
“You don’t understand!” whined Pettigrew. “He would have killed me, Sirius!” “THEN YOU SHOULD HAVE DIED!” roared Black. “DIED RATHER THAN BETRAY YOUR FRIENDS, AS WE WOULD HAVE DONE FOR YOU!”
What happened from Sirius' point of view. He convinced his best friend to use Peter as Secret Keeper as Sirius would have been the obvious choice in the eyes of their enemies, then he hears of the attack. He hurries there, seeing the house in ruins, pale and shaking, is comforted by Hagrid but Hagrid tells him he cannot have Harry. Then Sirius, obviously planning either to die or to go to Azkaban, leaves his motorbike behind to make Peter pay for what he's done, only for Peter to frame him, leaving everyone to thing it was Sirius who betrayed Lily and James.
Harry cries for the first time (ever, seeing he hadn't cried in the books yet even, aside from drying his eyes around Dumbledore and Lupin).
Made even worse that Harry practically wishes death on his godfather for something he didn't do which is, betraying and killing his parents.
Gary Oldman's portrayal of Sirius seems, for some, to be rather Narmy. However, when he screams "I DID MY WAITING! TWELVE YEARS OF IT! IN AZKABAN!", it becomes clear how broken he is.
Lupin's wolf form starts whimpering and giving Hermione an exceedingly pitiful look before fully succumbing to his feral state and howling furiously. It's almost as though, for a brief moment, the kindly teacher is aware of what he's become and truly regrets what he's about to do to his students.
Also before the transformation, Sirius is seen holding Lupin and anxiously asking if he took his medication and trying to soothe him. It feels like a good friend trying to help their own friend through a medical episode or attack, hoping it's not fatal.