The quiet moment shared by Edmund and Eustace when Edmund is the first to encounter Eustace after Aslan returns him to his human form - in particular, Eustace's attempt at apologizing for his behavior up to that point, and Edmund's reply:
"That's all right. Between ourselves, you haven't been as bad as I was on my first trip to Narnia. You were only an ass, but I was a traitor."
When the transformation of Eustace becomes known to the crew, Caspian and company are far more concerned with how inconvenient this is for them than with how Eustace must be feeling... except Reepicheep. The mouse spends his evenings keeping the unfortunate dragon company and telling him stories of other people who had suffered enchantments only to later have them removed. "It was meant well, and Eustace never forgot it."
The friendship that grows between Eustace and Reepicheep after the former is transformed into a dragon.
Drinian's minor freak-out when he learns that Reepicheep is the "man overboard". The Dawn Treader's captain, a seasoned, cool-headed, stalwart man, becomes nearly frantic upon discovering that his good friend might be in danger, despite his poor efforts to cover up his anxiety. Reep dousing Drinian with sea water by shaking his fur dry after being rescued is just the icing on the cake.
Fast-forward to Eustace's next adventure in Narnia. What's the first thing Caspian does when he see him? He gleefully asks if Eustace remembers how he broke Caspian's best sword on a sea monster. Decades have gone since that moment yet the king is able to recall it fondly.
The sacrificed slaves reuniting with their loved ones, with Gael and Rince diving over the side of the ship to swim out and meet her mother/his wife.
Lucy's friendship with Gael. Especially when Gael says "When I grow up, I want to be just like you," and Lucy, having worked through her own crisis of self-esteem (from comparing herself unfavorably to her older sister) replies "When you grow up, you should be just like you."
This simple line from Eustace at the end of the film. He and the other two hated each other at the start. Lucy especially says "how will we survive?" when they find out how long they'll have to stay. But going to Narnia helped bond them all together.
Eustace: We spoke often of Narnia in the days that followed. And when the war ended and my cousins went home, I missed them, with all my heart.
Even before that, where he apologizes for how he was before.
Eustace: Being a dragon wasn't all bad. (sheepish) I mean, I think I was a better dragon than I was a boy, really. I'm so sorry for being such a sop. Edmund:(sharing a grin with Caspian, along with Lucy) It's okay, Eustace. You were a pretty good dragon.
Edmund and Caspian's relationship in this version does seem more like close siblings rather than two different kings from different times.
Especially Heartwarming is Caspian handing Rhindon, the sword Peter had left with him, over to Edmund in the final act.
Caspian: Peter would have wanted you to have it
The look on Caspian's face when Drinian tells Edmund that Caspian is the one who gives orders on the Dawn Treader. Caspian clearly would not have minded sharing joint command with Lucy and Edmund, and is uncomfortable with repeating Edmund's orders word for word right in front of him.