The part at the end of DKC when Donkey and Diddy are at the treehouse having fun with each other is quite heartwarming.
Earlier, when they enter the banana hoard (that room that normally triggers the "failure" cutscene due to the entire stash of bananas being stolen) and they find their huge mountain of bananas returned to them.
When you 102% Donkey Kong Country 2, Cranky swallows his pride and admits he was wrong about Diddy not being cut out to be a game hero. He goes on to say he must take after himself and DK, subtly complimenting both of our favorite apes at once.
Also the cinematic for the game's 100% ending. Diddy and Dixie watch Crocodile Isle sink into the sea with the rescued DK, who has an arm around his little buddy and is carrying Dixie up on his shoulders. No spoken dialogue is needed to convey how proud he is of the younger kongs.
The ending has all four kongs celebrating by playing music together.
The fact Nintendo made a special announcement trailer for the portable and console trilogies returning to Virtual Consoles after mysteriously disappearing in 2010 shows a great deal of caring for fans; just look at the comments to see how many people are celebrating! They even retained Rare's logo in the ports, even though the company is now owned by Microsoft and Nintendo, as full rightsholder, could easily remove it.
The ending of "Speed"; DK is apologizing to Candy that his duties to protect the island always seem to get in the way of their dates. Candy, however, assures him that it's no big deal in song.
"Sometimes I feel like I'm in the way / But I know he's out there, saving the day"
When Skurvy and Klump are reunited in the show.
The episode "Speak No Evil, Dude"; DK is given a Sadistic Choice where he has to either stop the entire island from being blown up or save Diddy by finding him an antidote for a life threatening disease. DK resolves to somehow do both because, to him, the island isn't worth living on without his little buddy.
Dixie and Klump's friendship in "Klump's Lumps," and how Klump couldn't bring himself to betray her.
Just the fact that DK's nickname for Diddy is "little buddy," which is something taken right from the DKC manual.
The entire treasure hunt episode is one for DK. He initially refuses to go so he doesn't miss a date with Candy and only agrees when he realizes the treasure could be used to buy a fantastic present for her. And at the very end when the treasure is a dud:
Candy: Well, there goes my dream of buying the barrel factory.
DK: And there goes my dream of buying it for you. But who needs treasure when we've got each other?
(The two share their only uninterrupted kiss in the series.)