Heartwarming: The Legend of Zelda: Ocarina of Time
Mido is the resident bully in young Link's hometown, Kokiri Forest. After Link finds the Master Sword and travels seven years in the future where Ganondorf has taken over Hyrule, Mido guards the Forest Temple on Saria's wishes. When Saria is revealed to be a sage and leaves the forest for the temple of light, Mido asks the grown up Link (whom he no longer recognizes) to tell Link that Saria really liked him and that he's sorry for being mean to him. Awwwww....
There's a good chance that that Mido did realize that it was Link, and the whole thing was his way of saying sorry to Link while unable to fully admit that it was Link.
"I don't know why, but when I see you, you remind me of... him."
The final farewells between Link and each sage:
Saria: Saria will always be your friend.
Darunia: Remember: You and I are now true brothers!!
The Saria one even more so in the 3DS remake:
Saria: I will always be your friend.
Saving Lon Lon Ranch, Malon, and Epona from Ingo.
The dying soldier was mortally wounded trying to stop Ganondorf from capturing Zelda. Since Zelda escapes, he succeeded.
After the player defeats Ganondorf for the first time and Link and Zelda get out of the collapsing tower, Link and Zelda hear a noise from the rubble. Instead of jumping or gasping in surprise like any other person, Link's instinctive reaction is to throw his arm protectively in front of Zelda.
They also share a brief, relieved smile right before it happens. Not surprising, but how often do we see Link smile (especially this one) outside of Item Get?
The ending credits of Ocarina of Time, where all of the people of Hyrule gather at Lon Lon Ranch to celebrate the end of Ganondorf's regime. Highlights include Talon and Ingo drunkenly and happily leaning on each other like old friends, and the Cucco woman playing with her birds (ignoring her allergy to them). At the end, we pan to Death Mountain, where the Sages gather to look out in triumph over the land they have helped to save, with Saria riding piggyback on Darunia's back and Nabooru and Ruto sharing a significant look (with the implication that both pairs have become friends, based on their mutual love of the forest and attraction to Link respectively).
The first thing Link does when he returns to his original (young) timeline, is go and find Zelda in her castle. Cue THE END still-shot, complete with "Old Film" style screen coloring.
Robin Williams: The first time I saw you, I knew we would be linked forever. For you I traveled to the four corners of the world. I faced adversity. I became a hero.
Zelda Williams: Dad?
Robin Williams: I saved your kingdom!
Zelda Williams: Dad!
Robin Williams: ...yes, Zelda?
Zelda Williams: Are you mixing me up with the princess again?
Robin Williams: Hard to say, you're both pretty magical!
Many would consider it to be the Most Annoying Sound, but when you consider that Word of God compares Navi's feelings towards Link to how Tinker Bell feels about Peter Pan, the way Navi cries "Watch out!" when an enemy appears is just so... heartwarming. She genuinely cares about Link and wants him to be safe against the hordes of enemies he must battle.
Also when Navi helps you fight Ganon at the end.
Navi: There's no way he's going to hold me back again! This time, we fight together!
When you go to the Goron City for the first time as adult Link, you meet a young Goron who is Darunia's son, and is named after the legendary Goron hero. He is named after your character.