The one that stands out in my mind is Robb Stark reminding CatelynStark in A Storm Of Swords that Eddard Stark had four sons, not three. Thus reminding her that Jon Snow even existed. Shows how close the two brothers are. And throughout the book, Arya and Bran both remember their big brother, while in the previous two they forgot he existed. Even Sansa, who had followed her mother's lead in keeping her distance from Jon, realizes that she would really like to see him again.
Robb subsequently using his powers as king to legitimize Jon as a Stark and make him the heir to Winterfell, confidently assuring his mother that Jon would never harm a son of his.
King Robert Baratheon is normally a Large Ham and Boisterous Bruiser. However, when he first gets to Winterfell, the first thing he does is pay his respects to his lost love Lyanna Stark. Whether you view Robert as a good man or a bad one, his love for Lyanna and for the Starks as a whole, is genuine.
Willful, somewhat simpleminded, and lecherous Edmure Tully when his sister angrily demands to know why he let a bunch of useless people into a castle just about to be besieged: "Because they were my people, and they were afraid."
Jaime Lannister's kindness towards Pia, a physically and sexually abused peasant girl, is one of the most powerful moments of his Heel-Face Turn: He beheads one of his own men for raping her, and goes to great lengths to ensure that the budding relationship between her and his squire is a happy one. In a world where commoners, women, and above all commoner women are preyed upon by the likes of Gregor Clegane and Vargo Hoat, compassion like that really stands out.
When Lord Mormont gives Jon his family's sword.
Jon getting elected Lord-Commander of Night's Watch. For reference, Jon is the bastard son of Lord Eddard Stark, and was often derisively addressed by his comrades-at-arms as Lord Snow (Snow being a surname assigned to illegitimate children in the north). Only now, he really is Lord Snow.
Ned's conversation with Arya on her behaviour. "She had never loved him so much as she did in that instant."
Sansa Stark basically melts the heart of a drunken, scared, merciless and desperate (at that point) killer by singing a song to him.
The scene in which Sansa remembers a snowfight with her siblings and builds a snow-castle Winterfell. And then it is reversed when the only family she has left tries to kill her in the very same chapter.
Before he ruins thescene completely, Littlefinger is actually quite endearing when he stops to help Sansa to build snow-Winterfell and get's hit in the face when she throws a snowball at him. Helps that, judging by some of his facial expressions and actions, he genuinely seemed to be intent on helping Sansa build a snow castle by using his own creative mind. Makes you think that, despite all his actions as Littlefinger, Petyris still in theresomewhere...
Sansa hearing that her brother Jon has been elected Lord Commander of the Night's Watch, and wistfully thinking how good it would be to see him again; in previous books, when she thought of or referred to him at all, she was very cool and distant.
Jon turning down Stannis' offer of Winterfell and legitimacy out of loyalty to his father's Gods, his half-sister's claim of inheritance and his oath to the Night's Watch.
All the more so because it proves once and for all that Robb was right about him, and Catelyn was wrong.
What's more, he comes to the decision with the return of his direwolf Ghost, who had been gone for most of the book. He finally gets a moment of levity amongst the fighting and politics and it's Ghost's return which allows him to make his final decision.
Jon giving his little sister Arya the sword Needle as a parting gift early in the first book, as they are the closest siblings to each other (though Jon and Robb seem as close sometimes) and know that they probably won't see each other again for years. Plus, of course, "Stick them with the pointy end" and "Don't Tell Sansa!"
Arya throwing everything into the canals of Braavos except for Needle, because even if the Many-Faced God decreed that she could have no worldly possessions, the Old Gods are the ones in her heart.
Not to mention, Arya's thoughts on Needle in the same scene as above: "Needle was Robb and Bran and Rickon, her mother and her father, even Sansa. Needle was Winterfell's grey walls, and the laughter of its people. Needle was the summer snows, Old Nan's stories, the heart tree with its red leaves and scary face, the warm earthy smell of the glass gardens, the sound of the north wind rattling the shutters of her room. Needle was Jon Snow's smile." Needle is her family, the home she lost - Needle is Arya, and everything she ever loved, even when she's trying to be no one.
The Hound's Pet the Dog moment with Arya after the Red Wedding: "This thing about your mother.
Bronn names his bastard stepson "Tyrion," in a political climate where it would be impolitic to the point of suicide to do so. And gets away with it.
Thereby managing to combine Crowning moments of Heartwarming, Awesome and Funny in a single action. There's a reason he's an Ensemble Darkhorse.
And, honestly, the only thing likely to please Tyrion more than a baby named after him is a baby who was named after him to purposefully piss off Cersei.
Another Bronn moment is that after he refuses to be Tyrion's champion when he's on trial for murdering Joffrey, (because his opponent would be Gregor Clegane and Tyrion can't offer him enough to make it worth risking his life against such an opponent), Bronn still shows some concern for what will happen next. Bronn may be too much of a Combat Pragmatist to fight for Tyrion this time, but he still cares about him to some extent.
Bronn seems to have made Lollys Stokeworth his Morality Pet, being kind to her beyond the demands of necessity. Maybe the battle-hardened killer is willing to be nice to people who are no threat to him?
Well Lollys is a victim of a riot gang-rape and a bit daft but decent person, formerly single mother to his adopted bastard son, one of the last few living members of her family, not to mention his claim as head of the rich House Stokeworth. Perhaps despite what Bronn professes he has a soft spot for fundamentally decent/innocent people who suffer from some manner of social/physical handicap, are discriminated against by those around them and are caught up in bad situations not of their own design? Bonus points for those in some position of power who might help elevate his fortunes. Explains a lot, really.
Jaime: *THWACK* "You are speaking of a high-born lady, Ser. Call her by her name. Call her Brienne."
A subtle one, but Jaime's joke about only rescuing maidens after freeing Brienne from the Bloody Mummers is an understated way of making sure that she really was unharmed during her captivity. Considering that just a few chapters ago he was an amoral bastard who cared only about himself and his twin sister...
Also, "Her name is Brienne". What happened to calling her "wench", huh Jaime?
Ser Duncan the Tall, a lowborn Hedge Knight is to stand trial by combat for having attacked a member of the Targaryen royal family in order to protect a lowborn puppeteer woman who was being assaulted by the aforementioned Royal Brat. A large crowd gathers to watch the trial and Duncan wonders about why so many people would like to see his death. Then the smallfolk start issuing words of support, cheering for him, and some women even kiss his sword to wish him luck. He is in disbelief. "They are for me? Why? What am I to them?" His companions reply, "A knight who remembered his vows."
The North remembers and Ned's little girl... full stop... The love that the North still has for Ned Stark is touching and powerful indeed.
All the scenes with Penny, the cute little dwarf who pulls Tyrion back from the brink of Knight Templardom.
A small bit where the Tyrells are trying to wed Sansa to Willas Tyrell. Margaery sees Sansa's doubts and begins to speak lovingly of her brother's attributes. Keep in mind that they don't actually need Sansa's approval or consent; Margaery just wants her brother to be happy and for Sansa to see the same good in him that she does. And it worked, as Sansa was looking forward to the wedding. Too bad the Lannisters got wind of the plan.
Sansa daydreaming about what it would be like to be married to Willas; she imagines them with puppies in Highgarden, and thinks about the names she would give her sons: Eddard, Brandon, and Rickon. She even pictures a daughter that reminds her of Arya.
The entirety of Sansa and Margaery's friendship. Although the events of the story prevent them from keeping close and the fact that the Tyrells might be manipulating Sansa, Margaery does genuinely seem to care about Sansa and calls her sister. This is particularly shown when, despite cutting ties with Sansa when she is forced to marry Tyrion, Margaery still sheds tears for her and her situation.
The scene where Pod rescues Tyrion from Ser Mandon Moore. First because of Tyrion automatically assuming that it was Jaime, thus reminding us that only Jaime, Jaime, has ever treated him with kindness, and then for the subsequent revelation that it was plucky young Pod, the previously useless, timid squire, rescuing his liege.
Tyrion showing a newly disabled Bran how he can still learn to ride a horse.
After Jaime gives Oathkeeper to Brienne and sends her off to find and protect Sansa on his behalf (itself a minor moment), he turns to the White Book where the deeds of the Kingsguard are recorded (and he had been disappointed before to find his own list depressingly short because of the disdain Ser Barristan Selmy held for him—Selmy didn't even include most of his tournament victories). Struggling to write with his left hand, he records his defeats, imprisonments, disgraces and maiming. But he ends with "Returned safely to King's Landing by Brienne, the Maid of Tarth."
Even more so when you consider how much it would mean to Brienne, who only strives to live her life by knightly ideals even though she knows for the most part she'll get nothing but scorn for it, to know that a deed of hers has been recorded in the legendary White Book for all posterity. Gods I hope she finds out about it someday.
Jaime and Tyrion Lannister are neither of them sweet fuzzy bunnies. It's clear, however, that there is a great deal of genuine affection and trust between them. It's especially heartwarming because it tends to turn up the most at moments when each is at his lowest. When Jaime is trying to get back to King's Landing from Riverrun, for example, he keeps himself going by remembering that he's going home to Cercei AND Tyrion, and the first thing that the usually proud Tyrion says when Jaime sees his scar is, "well, they sent me into a battle without my big brother to protect me."
Throughout A Storm of Swords, Jaime seems to find Loras Tyrell arrogant and rather annoying. When he asks Loras about Renly, however, all of Loras's arrogance fades, and Jaime realizes that he was basically the same person when he was younger.
Tommen standing up for Margaery when Cersei threatens her. It shows that although he is far too young to love Margaery as a wife, he cares about her like a little brother would. Unfortunately, it doesn't end well for him.
One thing that makes these books frustrating is that so many of the reunions that we hope for never happen. One does, however: after more than a year apart, Jaime rescues Tyrion from the dungeon and the two brothers see one another for the first time since Ned Stark's feast. Fittingly for the two snarkiest characters in the series, it starts with a joke.
Tyrion: Come and get me, you sons of a poxy whore! Afraid to fight a dwarf?
Jaime: ... now, is that any way to speak of our lady mother?
A minor scene but once Bran mentions that the statues of the tombs in Winterfell are usually reserved only for the Stark kings and lords, it reminds us that Ned's deceased siblings, Brandon and Lyanna, have their own statues next to their father Rickard thanks to Ned insisting to have them carved. It really shows how much love Ned has for them as well as doubles as a Tear Jerker over the fact they have died rather young.
The moment in the first book when the Hound wins the tournament by saving Ser Loras' life, and the crowd cheers for him for the first time ever.
Tyrion gets a few Calling the Old Man Out moments before he and Tywin have their final reckoning but this one always stood out to me as a bittersweet one that might overlap a bit with Tear Jerker territory after Tywin asks if he'd forgotten that he's supposed to be impregnating his wife Sansa
“I had not forgotten, though I’d hoped you had. And when do you imagine Sansa will be at her most fertile?” Tyrion asked his father in tones that dripped acid. “Before or after I tell her how we murdered her mother and her brother?”
Tyrion seems rather protective of Sansa during their marriage, even though it was forced. After the Red Wedding while Joffrey is gloating about his victory and saying that he's going to make Sansa kiss her brother's severed head, Tyrion coldly hits him with the following:
Tyrion: Sansa is no longer yours to torment, you monster.
Though the scene is rather unsettling from Sansa's point of view, in hindsight Tyrion and Sansa's wedding night could be seen as this. Tyrion refuses to force himself on her and rather kindly tells her they can wait to consummate the marriage until she is ready. The whole situation sucks for both of them, but unlike the rest of the Lannisters, Tyrion at least seems to care about Sansa's feelings.
During the actual wedding, Garlan Tyrell reassuring Sansa that her husband is better than the rest of his family, praising Tyrion's actions and plans during the Battle of the Blackwater and saying he was "made to do great deeds". At least someone noticed all the good Tyrion has done.
In the beginning of Game of Thrones, when Tyrion is hanging out with the men of the Night's Watch, Maester Aemon calls Tyrion a "giant among men." Tyrion is almost too touched to say anything.
At the end of Daenerys's story in A Storm of Swords, Dany wakes up in the middle of the night with her servant Missandei, looking over Meereen for a house with a red door.
Oberyn Martell reminiscing about his visit to Casterly Rock with his sister, Elia. Elia picks up a baby Tyrion and coos over him.
I think it counts as 'Heartwarming - for a Lannister' that Tyrion has clearly known about Cersei and Jaime's romantic relationship for quite some time, yet has never used the information against them. Contrast with the nanoseconds that it took both Lord Tywin and Cersei to use knowledge of Tyrion's personal life to try to manipulate him.
In ADWD, Theon thinking about Robb: Theon: Where was I? I should have died with him.
After struggling with an identity crisis for the entire novel, and suffering through a horrible amount of torture, Theon managing to reunite with Asha, having regained some sanity and positive self-image.
Theon: Theon, my name's Theon. You have to remember your name.
Book 2 has the one where Stannis tells Davos that he is right for wrongfully putting his trust in his fickle lords bannermen.
After Tywin's funeral, he also confronts Cersei on how badly she has fucked up as a parent, vowing to raise Tommen properly after he is appointed Regent. At the end of book 5, he's shown having supper with Cersei and Tommen, admiring Tommen's good nature and kind heart. It makes his death at Varys' hands even sadder, although Varys at least comments that Kevan is a decent man, which (barely) softens the blow.
A moment that proves in a Lovable Rogue has his feelings: when Davos is returned to Dragonstone and taken to see Sallador Saan, the sly, swift-talking pirate stops still and stares at his friend, apparently back from the dead. There is a pause, and the normally-cheerful, dismissive Saan makes a remark about having something in his eye. So did we, Sallador. So did we.
After the events of The Red Wedding, no one could blame Edmure if he resented or hated Roslin for what the Freys did to the Starks and Tullys. But he doesn't. Not only does he never say an unkind word about her, but he still clearly considers her to be his wife, hopes to be reunited with her, and defends her when Jaime implies that she had any active role in the massacre. In spite of everything that happened, Edmure still loves Roslin enough to understand that she was an innocent pawn in what happened.
Hoster: "I saw. Last night, when it began, I told them...had to see. They carried me to the gatehouse...watched from the battlements. Ah, that was beautiful...the torches came in a wave, I could hear the cries floating across the river...sweet cries...when that siege tower went up, gods...would have died then, and glad, if only I could have seen you children first. Was it your boy who did it? Was it your Robb?”
Catelyn: "Yes, It was Robb...and Brynden. Your brother is here as well, my lord."