Every instance of Aang and Katara calling each other "sweetie". Ooginess aside, it really shows how far they've come since their first meeting.
Also the subtle nod to their Relationship Upgrade with Aang's new glider—which he and Katara fly on together—colored Water Tribe blue.
One heartwarming scene in The Promise part 2 is at the beginning, when Aang and Katara nestle up beside each other.
Sokka found Boomerang again, after losing it during the airship battle!
This moment from Part 2 was so heartwarming, it made Toph tear up:
Ho-Tun: Sifu Toph, you're the first person to believe that we can become more than what we are.
Suki telling Zuko she's worried about him; this guy burned down her village in Book 1 so it's great to see them on such good terms now.
In Part 3, Katara blows away every imaginable argument over whether or not she's suited to be Aang's "Forever Girl" by instinctively defining the heart of the conflict around them in a handful of sentences. Doubles as a Crowning Moment Of Awesome because doing so not only keeps Aang from repeating Roku's mistake and starting the war up again, but lays the foundation of The Legend of Korra's United Republic Of Nations.
Katara: On our first visit to Yu Dao, when I saw Kori's family—(a Firebender and his Earthbender wife Happily Married with an Earthbender daughter)— I also saw our future. (shows Katara and Aang in their twenties, with Katara holding a baby) If the nations have to be separate, what will that mean for us? I... I know it's selfish of me to think like this. There's so much more at stake than just us. Go find a quiet place, Aang, and figure this out. Then, whatever decision you make, I'll trust that it's the right one. I'll support you. Even if it means you have to fulfill your promise. And even if it means we have to be apart. (awesomely emotional kiss)
It's also a great marker of their Relationship Upgrade; in the penultimate episode of the series ("The Ember Island Players") she refused to confront her relationship with Aang during the culmination of the Hundred Year War, feeling that whatever was between her and Aang was not important when a more larger-scale issue was at hand.
When Aang admits that Zuko was right about Yu Dao; all that drama and they're still friends.
When Aang says that in a way Zuko is his family, being the great-grandson of his previous incarnation.
The fact that Aang's immediate response to his friend passing out from stress and exhaustion is to take him to his uncle Iroh. Then they have tea and talk.
Zuko saying that the moment he fell from the cliff, he knew Aang would save him.
Somewhere between heartwarming and a Tearjerker: by the end of the comic, Zuko started referring to Ozai by name rather than calling him his father, signifying that Zuko had finally managed to break Ozai's hold on him.
General How of the Earth Kingdom army gets a small one near the end. When the Fire Nation army arrived to defend Yu Dao, How urged Aang to kill Zuko and end hostilities before more lives could be lost. But when Aang tears Yu Dao away from the continent and Zuko falls into the resulting chasm, How grabs for him, trying to save his life.