In Wind Waker, Link's grandmother gets sick and delirious in the first half of the story. When you get back, she's begging you and your sister not to leave "your poor grandma all alone." If you cure her with a fairy note and really, why wouldn't you try?!, she gives a real tear jerker of a speech about how she'll try to be strong.
What makes it even sweeter is that it's a rare moment of Link having someone else show parental love for him. After so many games of him being pretty much alone to go risk his life in monster-infested dungeons and temples, how many times has he had someone say "I should be the one taking care of you"?
Even the practical, non-emotional reward for doing this is rather heartwarming: after you get Grandma all better, she whips you up a batch of your favorite home-made soup! (The fact that this is 2 helpings of the game's ultimate healing item simply makes this a Karmic Jackpot, and a just reward for your kindness.)
The entire ending of Wind Waker: from the King's Heroic Sacrifice, Hyrule's destruction, and finally Link setting off with the Pirates to find a new land everything fills you with a sense of hope for the future.
King: I have scattered the seeds... of the future...
Pretty much any scene with both Link and Tetra in it can be enough to make any gamer's stone heart turn all mushy...
The whole final fight. You were not fighting for Hyrule: all that was being destroyed around you. The only thing you were doing was fighting for your own lives so you could go find a new land.
How much Link cares for his little sister Aryll is really touching. Even the his facial expression and little laugh when he finally reaches Aryll in the cage is enough to make you all tingly.
The statue of the Hero of Time and the images of the 6 Sages in stained glass really drive home the awe and reverence that the people of Hyrule had for the people who had sacrificed so much, including their lives, to ensure that the kingdom was safe. It's nice to know that, with the Continuity Snarl that exists among the games at times, that at least once the actions of Link and his allies was remembered and honored.
Many of the interactions with Medli are just adorable. One particular moment in the Earth Temple:
Medli: I hope you're not offended by my suggestions. I just want to be helpful...
Honestly, learning that Ganondorf was just a Well-Intentioned Extremist who lost his way, only to finally look back and truly feel BAD for all that has happened. He may be evil, he may be the manifestation of Demise's rage and anger, but in the end, his whole reason for doing evil was for the people he had to rule. His final words after going on about how much he hated the wind really just seems to show final, inner peace, Ganondorf now free from all the evil he has done and finally able to rest.
Ganondorf: *Has just been stabbed through the forehead. He calmly smiles after a few seconds of Link standing on his head, the Master Sword shoved through him.* Ughnn....Heh heh...The wind...it is blowing... *Turns to stone as Hyrule drowns around him.*
Meta example from the HD version: That fact that people often send Pictographs of missable characters via Tingle Bottle to help others complete their Nintendo Gallery.