At the end of The Order of the Phoenix, after Sirius's death and probably knowing how bad Harry was feeling, several of the Order, including Moody, Lupin, Mr. Weasley, and Tonks, go up to the Dursleys. They all speak in casual tones to them, but the threat is still there. If the Dursleys mistreat Harry in any way, the Order will find out.
Given that this is Moody, who's a big softie at heart but will own you if necessary, makes it all the more heartwarming. Moody's probably smiling, off-page, at every threat Tonks makes to the Dursleys.
The fact that it's McGonagall saying the following quote is what makes it a million times more heartwarming than if anyone else had said it. And right in front and in full defiance of Umbridge, no less. And in the end, Harry does indeed become an Auror. McGonagall must have done a jig in her office when it was announced in the paper.
McGonagall: Potter, I will assist you to become an Auror if it's the last thing I do!
Order of the Phoenix is full of little ones of these. Like, after Percy sends Ron a letter telling him to stop being Harry's friend because Harry's a nutcase and it might ruin Ron's future job opportunities, Ron tears up the letter in fury, apparently humiliated to be related to someone like Percy. Also, when Harry unfairly accuses Hermione of debating his sanity with Lavender, she calmly tells him that actually, she told Lavender to "keep her fat mouth shut".
Another little one comes in Umbridge's first class, where she refers to them being taught by "dangerous half-breeds," to which Dean Thomas immediately snaps "If you mean Professor Lupin, he was the best we ever..."; the fact that Dean says it as opposed to, say, Harry, shows that the whole class, not just the Power Trio, were willing to stand up for Lupin despite the fact that he was a werewolf.
And then there's Hermione kissing Ron on the cheek for the first time to encourage him for his first Quidditch match. It doesn't help much, but the description of Ron touching the spot she kissed him in amazement is extremely sweet.
Weasley is our King, Weasley is our King, He didn't let the Quaffle in, Weasley is our King.
Weasley can save anything, He never leaves a single ring, That's why Gryffindors all sing: Weasley is our King.
Hagrid's whole relationship with Grawp. "HERMY! WHERE HAGGER?!" He remembered Hermione's name. To clarify, everyone, everyone, from the centaurs to his three best friends, thinks Hagrid is insane for dragging Grawp back and keeping him in the forest, while Hagrid just insists that he has to because he's his brother. He gets beaten up, risks being fired, even has Hermione briefly siding with Umbridge against him, but in the end, not only is he vindicated when Grawp grows more gentle and docile, but it was his devotion to his brother despite all odds that meant that Voldemort wasn't the only one fielding giants in the final battle, which gave the defenders much more of a fighting chance than they would have had otherwise.
The fact that Grawp remembered Hermione's name.
A rather small one in Order of the Phoenix, when Umbridge fires Professor Trelawny and gleefully tries to throw her out, Professor McGonagall, who has previously made no secret of her disapproval of Trelawny's talents and teaching methods, defies Umbridge to rush forward and comfort the despondent Trelawny.
It's not specifically stated in The Order of the Phoenix, but Dumbledore smiled and was probably very touched when he saw that the students' resistance group was called Dumbledore's Army. Gains extra resonance after the Book 7's revelations. Remember, in his youth, Albus Dumbledore was preparing to lead an army of wizards to do exactly what Fudge was accusing him of — Take Over the World. Since then he's been deliberately avoiding power because he knows it will cause him to turn to The Dark Side. Now here is an army of wizards, declaring their trust in his humility and wisdom.
Two Mrs. Weasley moments in The Order of the Phoenix. The first is her boggart transforming into a dead Harry, whose death she is deathly afraid of, as indicated by the boggart, despite him not being part of her immediate family. Then, there is the other scene:
"He's not your son, Molly." "He's as good as!"
Harry's reaction adds to this effect.
More so is Sirius's. He just comes in the room and stares at the place where Harry's body was before Lupin vanished it. Molly's collapsed, sobbing all over Lupin, and Sirius is just staring at the place where Harry was. You can totally imagine what he's thinking, how terrible and grief-stricken he would be if it happened, and it makes me want to just cry my heart out. Also Sirius comforting Molly, who were at loggerheads in who loves Harry more, telling her that regardless of anything the Order are True Companions and that if anything happened to her, they will look after her sons.
Also, Sirius isn't just seeing Harry; the eyes of the dead boggart are shut, so physically, Sirius is seeing James, his best friend, lying there dead.
There's one tiny moment that's incredibly heartwarming for how subtle it is. After seeing Ron make a fool of himself, basking in his new found glory as a Quidditch hero, Harry smiles and the lines note that Harry was reminded of his father. This is heartwarming because Harry suffered a Broken Pedestal on seeing his father bullying Snape and acting different from how he imagined and being unconvinced by Sirius and Remus telling him that his Dad was fifteen. Seeing Ron behave like that, makes him understand those words for the first time and love both his father and Ron.
A small one: When Neville is nervous about his Transfiguration O.W.L., Professor McGonagall says there's nothing wrong with his work except lack of confidence. She may be a Stern Teacher, but she will encourage her students when they need it.
Though doubling as a Tear Jerker , Neville visiting his parents, who are near catatonic and don't seem aware of the world around them. His mother gives him a gum wrapper, but instead of throwing it away Neville keeps it.
The teacher's reactions to Harry's interview. Professor Sprout awards him twenty points for passing her a watering can, Professor Flitwick gives him a box of Sugar Mice and Professor Trelawny changes her usual prediction of Harry's death by predicting he will live long, have twelve children and become Minister of Magic.
From the Film:
Harry and Sirius hugging when Sirius greets Harry.
Harry is feeling both depressed and scared that he might be becoming something that he isn't, that he is becoming more like Voldemort.
Harry: What if I'm becoming bad? Sirius: I want you to listen to me very carefully, Harry. You're not a bad person. You're a very good person, who bad things have happened to.
Any and all scenes where they hug.
Sirius: When all this is over we'll be a proper family. You'll see.
Which makes Sirius's death all the more tragic.
Order of the Phoenix has this:
Neville: Fourteen years ago, a Death Eater named Bellatrix Lestrange used the Crutiatus Curse on my parents. She tortured them for information, but they never gave in.(pause) I'm quite proud to be their son. But I'm not sure I'm ready for everyone to know just yet. Harry: We're going to make them proud, Neville. That's a promise.
Harry: You're the weak one. And you'll never know love or friendship. And I feel sorry for you.
The fact that Moody ushers Luna past him and gives her to Tonks in the battle of the Ministry; Tonks is clumsy and barely able to stand on her own two feet, but Moody knows that there is power inside Tonks and that she'll fight to the death to keep Luna and the others safe. The fact that he trusts her, a junior Auror, is enough to make this troper squee happily.
The part where Fred and George comfort the boy who just got out of Umbridge's torturous detention in Order of the Phoenix.
As Dumbledore's Army's fight with the Death Eaters in the Ministry is joined by members of the Order of the Phoenix, Harry defies Sirius telling him to run to fight by his side against Bellatrix, to Sirius' grudging approval. Harry gets off an impressive spell and Sirius calls out "Nice one, James!" The man who spent 12 years in a torturous prison, falsely accused of not just killing a dozen muggles, but also leading to the demise of his closest friends, who's mind is truly fractured, however well he hides it, forgets, for just a moment, that it's Harry fighting by his side, and thinks it's his old friend, James, fighting the Death Eaters just like old times. As much as people always say Harry has Lily's eyes, he is truly his fathers' son.
During the Dumbledore's Army scenes where Neville starts with absolutely no confidence whatsoever, unable to do something even as basic as Expelliarmus, but never gives up. When he does eventually pull it off, the entire class gathers around him to congratulate him. Even the person he disarmed is thrilled to see him succeed. It doubles as a Crowning Moment of Awesome for Neville as well because he never gave up, not even against his own lack of self-esteem.