Two Harry-Dumbledore moments that positively ache occur in Half-Blood Prince and Deathly Hallows:
"I am not worried, Harry. I am with you." "Of course this is all in your head, Harry, but why on Earth should that mean it isn't real?"
Two words: Alice Longbottom.
Just to expand this, Alice and her husband were tortured to madness for information by the Death Eaters. In Order of the Phoenix, the gang visit St. Mungo's Hospital after Ron's dad Arthur is bitten by Nagini, Voldemort's snake. Inside, they come in a room where they meet Neville (who did not come with them) and his grandmother. They meet Neville's parents, (whose minds are damaged beyond repair), and Alice, Neville's mother, gives her son a candy wrapper, meaning that after all these years and despite her madness, she still remembers her son and still loves him.
Furthered when Neville's grandmother tells him to throw the candy wrapper away ("She must have given you enough to paper your bedroom with,"), but Harry catches Neville slipping the wrapper in his pocket.
The bit that really clinches this as a moment of Heartwarming is Neville looking at his friends, daring them to speak. He's willing to fight for them, there and then, if his friends dare say anything.
Snape's worst memory is his falling out with his best friend. Nothing in all 38 years of his life was as bad as that to him.
Not just his best friend, the girl he had been in love with from childhood. Think about it - up until that point, Snape stood as much a chance as James did of ending up with Lily, if not a better chance. That would be anyone's worst memory, I think.
Going back to the first book, there's the beginning of Harry and Ron's friendship with Hermione, when they go back to save her and she winds up lying to protect them. It's really all summed up with the classic line, "There are some things you can't share without ending up liking each other, and knocking out a twelve-foot mountain troll is one of them."
One of the best things in the series is the friendship of Harry, Ron, and Hermione. I can't really think of any moments involving Hermione that haven't been mentioned, but there's two with Harry and Ron that make me "aww" every time. The first is in The Goblet of Fire, right after Harry wins the first task. Ron and Harry have been fighting, but Ron comes back after this, and immediately starts complaining that Harry should have gotten more points for fighting the dragon. Harry thinks to himself that Ron's friendship is worth all the points in the world. The second is in The Deathly Hallows when Ron returns after running off, telling the story of how the Deluminator led him back. He says, "I guess Dumbledore always knew I'd leave." Harry says, "No, he knew you'd always want to come back."
The development of Harry's friendship with Luna. In the Fifth book, he is embarrassed to be seen with her, but he's grateful that she believes him about Voldemort's return and sympathizes with her tragic experiences. By the sixth book, he is highly offended when it's suggested that Luna isn't worth his time, and actually takes her to Slughorn's Party. For whatever reason, Harry and Luna have this empathetic connection that he has with no one else - including Ron and Hermione. Harry comes to appreciate her indestructible sense of optimism and her careless honesty, which, considering how half of his life seems to be secrets and confusion, is probably refreshing. There's a reason Harry gave his daughter the middle name "Luna".
It's more than that; while Hermione and Ron are his friends, Luna can sympathise with him; like Harry, she has lost a parent - she knows what it is like to lose a mother. Neither Hermione nor Ron do, but Luna does; she is the one person who can really and truly understand Harry's plight.
To finish off the series, Jo writes in her dedication: "...and to you, if you have stuck with Harry until the very end." A touching nod to all the HP fans — many of whom did start the series as children right around Harry's age and finished it as adults. Personally, as eager as I was to dive headfirst into the final book, that little acknowledgment gave me pause to reflect that after all this time it really was the end, guys!
Molly Weasley's entire character. In contrast to Aunt Petunia, who was related to Harry by blood and given the responsibility of caring for him but could not muster so much as a drop of affection for him - even at the very end when she knew he deserved better - Molly was not related to Harry by blood, willingly took on the responsibility of caring for him, and loved him as a mother would a son. And all this despite having seven biological children of her own. It really speaks volumes to the size of her heart. But Good Is Not Soft. When Bellatrix threatened her only daughter, it didn't matter that Molly was going up against a mass-murdering Ax-Crazy psychopath despite not having seen much battle in twenty years, she jumped directly into the fray and ended that bitch because, by God, if she's around, you will not lay a hand on her children. She might not have made a story-altering Heroic Sacrifice like Lily Potter did, but she touched Harry's life forever by living the way that Lily died. And if the idea of that doesn't warm your heart just a little, then what will?
Meta-example; "When I’m 80 years old and sitting in my rocking chair, I’ll be reading Harry Potter. And my family will say to me, “After all this time?” And I will say, “Always.”" — Alan Rickman
At the premiere of The Deathly Hallows — Part 2, Jo Rowling had this to say: "The stories we love do live in us forever. So whether you come back by page or by the big screen, Hogwarts will always be there to welcome you home." And she was right.
And, she was almost crying as the crowd chanted: "THANK YOU! THANK YOU!" repeatedly.
Pottermore (Warning, Spoilers for Book 7)
Harry and Viktor Krum meeting before the final of the 2014 Quidditch World Cup and embracing each other like old friends.
Harry also introducing James and Albus to him.
The Reveal that Harry and Ginny named Neville Albus' godfather.
Ginny and Harry not minding that Albus cheers for another team. Good Parents indeed.
Krum winning the tournament and being hoisted into the air by his team and having "tears" in his eyes.