In Harry Potter and the Goblet of Fire, as the realization of Cedric's death is sinking in, and Harry is surrounded by people but doesn't want to be seen crying:
Mrs. Weasley set the potion down on the bedside cabinet, bent down, and put her arms around Harry. He had no memory of ever being hugged like this, as though by a mother.
Harry: I told him to take the cup with me. (eyes and throat burn)
Also, after being named the Fourth Champion, nearly everyone — including Ron, Harry's best friend — believed that Harry had somehow found a way to sneak into the tournament, just to gain fame. However, the next morning, before demanding answers from Harry or even mentioning the whole fiasco, Hermione brought Harry a piece of toast, knowing he didn't want to eat in the Great Hall with all the attention he was getting, and asked him if he wanted to go for a walk. Harry later told her everything that happened and "to his immense relief, Hermione accepted his story without question," as quoted from the book.
A rather minor one, but in Goblet of Fire, the fact that Harry takes it upon himself to warn Cedric about the dragons in the First Task to be an immensely heartwarming simple gesture of genuine, selfless decency. It's moments like that just as much as — or more than — the huge battles which make Harry a worthwhile hero.
Likewise, the conversation leading up to Harry and Cedric grabbing the Cup is among the most heartwarming moments of the series. Cedric, who was so popular during the tournament, giving up glory to Harry, who had been tormented all year long. Harry, who had held a serious grudge against Cedric because of his relationship with Cho, suggesting that they take it together. It was a beautiful moment — and it made what happened next that much more shocking and terrible.
It's even more poignant because, as it's pointed out in the book, Cedric is willingly giving up the kind of glory Hufflepuff hasn't even come close to in decades. As is so often seen in the fandom, Hufflepuff has an (undeserved) reputation as the useless house, the "load o' duffers", the ones who aren't special or important in any way. Cedric knows that. He knows that if he and Harry walk out of the maze both holding the Cup, it won't be Hufflepuff that goes down in history; it will be Harry Potter, once again. He willingly gives up a chance to singlehandedly change Hufflepuff's reputation forever, solely because it was the right thing to do. He would rather give up the glory altogether than take a victory he doesn't feel he earned—He and Harry both saved each other in the maze, they both earned the victory, and by god they're both going to share in it.
A very small, subtle one in The Goblet of Fire. Harry has rescued Ron and Gabrielle from the Lake when "[...]Percy, who looked very white and somehow much younger than usual, came splashing out to meet them. [...] Percy seized Ron and was dragging him back to the bank ("Gerroff, Percy, I'm all right!")" Percy, who has always been seen as pompous and annoying, caring more about his job than his family, was plainly terrified for his little brother despite knowing Dumbledore would never have let anything happen to him.
While the entire scene where Dumbledore, Hermione, and Harry confront Hagrid after he is outed as Half-Giant by Rita Skeeter is this, the part where Dumbledore casually mentions that many parents have written letters in Hagrid's defense is touching. It's not just that his friends love him, it's that many people who have known him do so as well that just makes it that much more awesome.
The strength and hope that Harry and co take home with them at the end of the year (particularly in the film), even though the entire wizarding world is about to enter into utter turmoil.
If it hadn't been fake and self-serving, Moody comforting a distraught Neville after showing off the cruciatus curse in lessons, as Neville has just seen exactly what happened to his parents.
From the Film:
In Goblet of Fire, Dumbledore talking to Harry after the end of the end of year feast ("Remember Cedric") about how he has friends at Hogwarts and isn't alone... then lightly brushes his cheek with his hand on his way out of the room. Dumbledore was rather flamboyant in the film, but that bit right there... *Sigh* He really does love him.
In The Goblet of Fire, they're all being taught to dance for the Yule Ball, and while the girls are all eager, none of the boys leave their seats. Then you see Neville hesitate (clearly trying to work up the nerve) before getting up to dance. Later, we see Harry and Ron entering the Gryffindor dormitory to find Neville, in his pyjamas and dancing shoes, practicing the waltz by himself, humming a tune as he goes. Even later than that, after the party, Neville comes back later than everyone else, still humming and twirling round. The fact that they removed the part about him first asking Hermione and being taken by Ginny as a sympathy date makes the whole thing really sweet, as it implies that he was genuinely excited about going with Ginny and that even a nerd like him could get a date.
Fleur's reaction to Harry having saved her sister "even though she wasn't yours to save" really serves to humanize her character; Clemence Poesy did a wonderful job in that one small moment.
McGonagall gets three in one scene. She's less than a footfall behind Dumbledore as they burst in on Harry and Moody, she holds an arm out protectively when Harry passes her and she is the one to hold Barty Crouch Jr at wand's length when he lunges for Harry.