And, of course, a very simple screen with nothing more than ordinary text on a blank background, that makes it all worth it: Dawn of A New Day.
Link burying Mikau after healing his soul. Even though they had just met and Link has other people to help, he was kind enough to give him a proper burial.
Romani ranch. If you fail to save Romani on the first night, she becomes a semi-mute, heavily traumatized pseudo-zombie when she is returned a little while later. However, completing the entire sidequest and coming back on the final night reveals that Cremia is giving her younger sister a Romani Mask to signify her becoming an adult. Romani is excited, but Cremia is morose because she realises the world is about to end. The last words Cremia says to her sister are, "Romani... sleep with me in my bed tonight, OK?" A CMOH and a Tear Jerker all in one.
Also, the scene where Cremia is giving Link a ride to town. Particularly:
In town...I have a friend. Her name's Anju...The day after tomorrow is her wedding.
After you've gotten Romani's Mask, when you go back and defend the delivery again, Cremia either gives link a Huge (200) rupee, or, a hug. And we all laugh at the hug for what it ends up being. But think about how many times Link has ever been hugged. This is likely the first one he's received in quite a while: Hero of Time link grew up with the Kokiri, who are perpetually young and mostly innocent (i.e. not the affectionate type) and was ostracized by nearly everyone until he left. He spent a good deal of time getting abused as a child in Hyrule (and when offered a Goron hug he seems genuinely terrified of being crushed and runs away screaming), and even as an adult he only got to know girls on a temporary basis before they became Sages. All his (likely painful) exploits as an adult were lifted clean out of the stream of (this) history, including his relationship with Zelda. On coming to Termina, it's possible all physical contact he's even known has been painful. When he finally gets a hug, getting used to a warm and fuzzy feeling has nothing to do with being on the cusp of puberty; this is quite possibly the first time in his life Link has ever felt physical affection. The fact that his face is pressed into her boobs is just a bonus.
The resolution of the Cremia and Romani sidequest is particularly heartwarming and sad if only for Romani's reaction. She seems so happy to be getting a Romani mask and a Chateau Romani that it's really motivating.
Removing the curse on Pamela's father. The kicker to that scene is that if you try fighting him before healing him, you're treated to a cut scene of Pamela running in to stop you, and then begging her father to remember her. He crawls back into the closet, and she practically screams at Link to forget everything he's seen and leave.
The Postman sidequest. It all revolves around the postman and his inability to flee the city because it's not on his schedule. You can make it so that hours before the end, he is ordered to disregard the mailing schedule and flee the city. Upon receiving the news, he happily proclaims his freedom to "make his own schedule!", removes his hat, claiming "I don't need this anymore...", and runs out of town. In the process, he becomes what he was most remembered for in Ocarina of Time: that carefree guy that loved to run with no destination or goal in mind, but for the joy that he found in running itself. Indeed, in the credits, we see him exactly where he left off in the last game, aimlessly running in a large, open field, because even in a different dimension, that's the way it's supposed to be.
The cutscene where you learn the rest of the Goron Lullaby, especially the end. And then this:
"It's just like...daddy's...right...beside me..." (falls asleep)
A LOT sadder when you consider that his dad is probably dead because it's so cold out there... until you go back after beating Goht and he turns out to be alive and well. Then he tries to persuade you to become chief, which is a CMoH all by itself once all the Gorons start cheering for "Darmani". Especially when you remember the real Darmani's cutscene, and know it's just what he would have wanted.
Actually, his dad is the one who teaches you the intro to the Lullaby. You find him frozen outside, because the old Goron is so determined to go end the winter himself, for his son and for his people. It's a bit of a Tearjerker that he either can't remember the last notes of the song, or he's just too cold to play them (likely the former).
The defeat of Skull Keeta: After you beat him in a rather challenging challenge, he grants you his "soul" (the captain's hat mask), begs you to tell his men the war is over, then asks "Orders, captain?" Cue silence as you think a Song of Healing prompt is coming up. Instead, Link stands there for a second, then dismisses the captain with a slow, heartfelt salute. "Understood! Thank you captain!"
And when we say, dismiss, we mean dismiss from his eternal serving as skeletal captain. You can do the same to any skeleton soldiers you find by wearing the captain's hat. They're even disappointed if you don't dismiss them.
When Tatl chose to leave her brother Tael behind and follow you into the Moon to fight Majora's Mask instead of just leaving you once she and Tael were together again like she said when you first met her.
When speaking to the owner of the milk bar on the final hours (with the Romani mask, without it his non-mask dialogue isn't so heartwarming), he tells you that he refused to leave because, even if it will be his undoing, he's the kind of guy who will stay with his business in the end. Then he adds that he also stayed out of the faint hope his favorite customers will appear. Cue a ping to the heart when he says "And I wasn't wrong. See? You stepped in." Much more heartwarming when you've stopped by the bar right after reuniting Anju and Kafei, and adds more awesome to it right before you set off to kick demonic mask-ass.
More or less any person you add to your Bomber's Notebook can be this. In addition to the above, you can do little things to cheer people up like giving a nameless man paper to wipe his ass with, or helping a man whose only wish is to watch his chickens grow up to their prime by accelerating their growth into roosters.
Getting the couple's mask and showing it to the quarreling people in the mayor's office. They are reminded of young love, and stop fighting immediately. Also, if you show it to the traveling mask salesman, he remarks that it is filled with love, and that you must have gone to a lot of trouble to get it (And you did!) and that he wishes that he was looking for the mask so your efforts would amount to a little more than nothing. (As far as he knew)
If you help Gorman in the first day, he doesn't return to the bar in the second day, choosing instead to play cards with the juggling brothers.
The extra use of the Troupe Leader's Mask in the remake, where the older and younger Gorman brothers show their soft sides by making a hangover cure for their middle sibling.
Marching the Cucco chicks and helping them grow into roosters.
The end credits. A giant compilation of heartwarming moments depicting the happy resolve of everything you saw during the game.
The moon finally goes away and the soldiers rejoice.
The giants make peace with the skull kid.
The carnival begins.
The Zoras perform their much-awaited concert.
Great Bay is calm, with clear waters.
The spirits of Ikana are at rest, except for the ever-quarreling royal guards.
The monkeys and the Dekus live together in peace.
The postman cheerfully returns to his work.
Kamaro's dance lives on through the Rosa sisters and their creepy soldier-stalker guy.
Romani continues her archery with Cremia, who cheers her on.
The bomb shop owner and his mom leisurely sit on their doorstep.
Grog is surrounded by his crested, feathered friends.
Gorman gets his wish of seeing his beloved song performed by the Zoras.
Soldiers, Zoras and salesmen gather to hear Guru-guru's playing.
Pamela and her father are a happy family once more.
Anju and Kafei's friends and family come together for their wedding.
...Tear Jerker as the Deku butler cries over his son's remains and
Link leaves Termina and notices a quaint carving on a tree stump.