Heartwarming / Catching Fire

• Early in the book, Katniss says that she’s been trying to improve her relationship with her mother as her time in the arena made her realise that she cannot continue to be angry with her over her being negligent, out of depression, in the past.
• Katniss has an in-universe one shortly after being threatened by Snow and desperately trying to make him and Panem believe they are in love to avoid a civil war:
  we fall into the snow, me on top of him, and that’s where we have our first kiss in months. It’s full of fur and snowflakes and lipstick, but underneath all that, I can feel the steadiness that Peeta brings to everything. And I know I’m not alone. As badly as I have hurt him, he won’t expose me in front of the cameras. Won’t condemn me with a halfhearted kiss. He’s still looking out for me. Just as he did in the arena. Somehow the thought makes me want to cry.
• On the Victory Tour, Peeta pledges that he and Katniss will each give up one of their monthly parcels of food per year to give to Rue and Thresh's families to thank them for, respectively, being Katniss' friend and sparing her life. Peeta outright says that by saving Katniss' life, Thresh saved his, too. A single monthly parcel is noted to be enough to feed an average family for a year in the Districts. Katniss reflects on how Haymitch told her she "could do worse" than Peeta, which she already agreed was true—but in this moment, she thinks that she couldn't possibly do better than him.
• Katniss, when considering fleeing into the wilderness, doesn't just intend to get herself and her family out of the line of fire. She also wants to take Peeta, Gale, and Haymitch. No One Gets Left Behind.
• Katniss meeting Twill and Bonnie in the woods and giving them food. Bonnie's clearly never had that much food to herself in her whole life—and it's one bun, it's not that exorbitant. It reminds Katniss of the days when she was starving as a child, but it also feels good to watch Bonnie's overjoyed reaction to eating her fill for once.
• Peeta discovered which of his baked goods was Katniss's favorite—the buns with cheese baked into the top—and quietly kept them in constant supply in her cupboard.
• Katniss' prep team meeting her mother: they're actually quite nice to her and praise the way she styled her daughter's hair for her Reaping Day. Then, she proposes to show them how she did it and they happily accept.
  • The way this itself plays into Katniss' realisation of the book's key theme about who the real enemy is: Katniss immediately feels bad about thinking badly of her prep team and thinks about how, if she'd been born in the Capitol, it's very likely that she'd be a lot like them.
• Every night Katniss and Peeta sleep in each other's arms. Whether you view it as romantic or platonic it's still pretty heartwarming.
  • Peeta is a teenager, aged 16 or 17 during the events of CF, and with that age comes hormones... Every night he shares a bed with the person he's been in love with for 2/3 of his life, and they don't just lie side by side to have each other's presence. Katniss sleeps with Peeta as her pillow, her head resting right over his heart. The fact that he never once tries to make a move, suggest that they could kiss or caress, or even make any form of mentioning of his desire for her, or what two people can do in a bed together, is quite heartwarming (and, frankly, rather impressive). He doesn't want to make her uncomfortable; he knows how much it means to her to have him there during the night and he doesn't want to spoil that.
• Haymitch's summation of Katniss and Peeta's relationship:
  Haymitch: You could live a hundred lifetimes and never deserve that boy.
• Katniss noting that she doesn't want Haymitch to die in the arena in order for Peeta to live, because she views Haymitch as her family.
• It is quite cute about how Effie has become kind of a doting mother or aunt-type figure to Katniss and Peeta in Catching Fire. Especially seen when she tries to take them down to training, but Haymitch doesn't want them to look like they need a babysitter, so she just walks them to the elevator, fixes their hair, and pushes the button for them. It's possible that almost anyone who has a mother/guardian like that but isn't a little kid anymore can relate to the situation. Considering Katniss and Peeta's respective mothers, Effie (with all of her faults of being indoctrinated by the Capitol) is someone those two can rely on.
• Peeta's showing of skill during the Quarter Quell, The painting of Rue.
• Countless moments between Katniss and Peeta in the book, frankly. Examples include the pearl scene, Katniss' despair when she thinks Peeta is dead, the moment they share at the beach, the day they spend at the rooftop together...
  Peeta: I just want to spend every possible moment of the rest of my life with you.
  Katniss: Come on then.
• "I'll allow it."
• "With my paint box at home, I can make every color imaginable..."
• When Peeta mentions that nobody really needs him, Katniss finally realizes that she cannot live without him. At this point you know that she loves him for real.
  "No one really needs me,” he says, and there's no self-pity in his voice. It's true his family doesn't need him. They will mourn him, as will a handful of friends. But they will get on. Even Haymitch, with the help of a lot of white liquor, will get on. I realize only one person will be damaged beyond repair if Peeta dies. Me.
  "I do," I say. "I need you."
• Finnick Odair. He's built up by Katniss as both an irredeemable manslut and a very dangerous enemy who won his Games when his sponsors sent him a weapon and he killed everyone in his path. As Katniss (and the reader) get to know him, however, he becomes charming, funny, tender, caring, heroic, pitiful, tragic, and sacrificing. He still has no qualms killing in self-defense, having gotten first blood in the Quarter Quell, but he plainly cares deeply for Annie, Mags, Katniss, Peeta, and Johanna.
• Mags volunteering to take Annie's place. Katniss understands later that she could not possibly have believed she would survive at her advanced age, but nevertheless did it to protect someone. This also inverts the stereotypes involved with the Career Districts and their volunteers: instead of volunteering for glory and riches, she did it as a sacrifice like Katniss did a year prior.