Haymitch trying to comfort Katniss after she sees the old man in District 11 getting shot.
Cinna seeing Katniss again:
Cinna: Are you okay?
Cinna: Ready to work?
Cinna: All right. Good.
Effie completely loses her Capitol veneer after the announcement of the 'twist' for the Third Quarter Quell. Suddenly, between her tearful performance during the Reaping, her insistence that Peeta and Haymitch get something gold to match Katniss's mockingjay pin and her hair ("We are a team!") and her heartbroken goodbyes to Katniss and Peeta ("You both deserved... so much better..."), it's clear that she loves her Tributes just as much as anyone else does. It's both sweet and tear-inducing.
It's almost definitely tear-inducing, since when she first met them, they were just another round of Tributes that (considering Haymitch was the only previous Victor from 12) were going off to die. Then they not only won, they both won; she reacted much more like someone whose pumpkin just won best in show, they were glamorous trophies, "her" victors. It wasn't until they spent most of the next year together, getting to know each other, then had the Quarter Quell announced that they were going in once more, almost definitely due to their actions, when she realizes she's not just the manager of Panem's latest media showcase, they really are a team, and she really does care about them.
That famous line after Katniss has a nightmare on the train, and Peeta comes in in a panic at the sound of her screaming.
Katniss: I... I'm sorry... I had a nightmare...
Peeta: I— no, no, it's okay. I... I get them too. I'll, um—
Katniss: Wait! W-wait... please... stay with me?
Peeta:[astounded] Yeah... yeah of course. [walks over and holds her] Always.
They always lay together in the same position—arms around each other and her head on his chest. This was the exact position they were in while in the cave during the first movie's Hunger Games. From the book, emphasis added:
We manage the darkness as we did in the arena, wrapped in each other's arms...
Katniss clearly falling for Peeta during their time on the beach.
Him finding the little black pearl in the oyster and handing it to her.note Ironically, this was a very humorous moment in the book because Katniss reminds Peeta of when Effie was mistaken about the origins of pearls.
Their [arguably] first real kiss after Peeta insists she has more to live for.
Peeta: Nobody needs me. Katniss: ... I need you.
The fact that the trinket Peeta had Effie make for him was really a locket for Katniss with a picture of her family and Gale.
Katniss's obvious relief after Peeta hits the force field and almost dies. You can see how much she loves him.
Peeta:[gasping for breath] Wa... watch out, there's a forcefield there...
Katniss:[openly sobbing and hugging him] Oh my god, Peeta! You were dead! You were dead! Your heart stopped!
Peeta:[smiling and hugging her back] Don't worry, don't worry... it's working now.
Katniss standing in front of the Peacekeeper whipping Gale and sitting by him the entire time he's healing.
Peeta coming over to look after Gale so that Katniss can get some sleep. It must be incredibly painful for him since it's implied to be the point where he begins to feel convinced that she loves Gale, and all the same he comes over to help her and look after his romantic rival. note In the book this scene is followed by Katniss going up to bed and wishing Peeta was there to hold her, thus rather cementing the fact that she's falling for Peeta and her feelings for Gale are platonic.
Haymitch all but shoving Peeta behind him when they interfere with the whipping, so if Thread kills anyone, he'll kill Haymitch first.
Katniss bonding with Mags and even offering to teach her archery is pretty sweet as well.
Mags' whole relationship with Finnick. They don't even have to speak for you to feel how much they care for each other.