The end of the flashback sequence between the Figaro brothers. "To Figaro!"
The moment when Setzer, following a reflection on his relationship with Darill and her disappearance, boards Darill's airship with Celes and the Figaro brothers in tow, kicking off "Searching for Friends" as the new overworld theme, giving a much needed reprieve to the despair resulting from Kefka's destruction from the world.
The scene in Mobliz where Terra finally rejoins the party. Turns out the orphans don't care if Terra is an Esper or a monster. She's still their Mama. And then there is Terra's cry of "I know what it means to love!"
Which segues nicely into a Moment of Awesome where she single-handedly does what your entire team could not (which is at least two others).
This is even more heartwarming if you remember her line ("I'm not some love-starved twit!") from earlier in the game right after she joins your party, and realize how much she's defrosted since that point.
Especially at the end of the scene in Figaro Castle when you take them both there, and you learn just how much Edgar loves his younger brother: literally throwing away his own chance at freedom by rigging a coin toss against himself, taking on the entirety of the burden neither of them wanted to bear and giving Sabin the freedom to choose his own path in life.
It is impossible not to shed at least one sentimental tear during the song Celes sings in the opera scene... or any time that song is reprised, really.
It's almost hard NOT to. It's HER theme tune.
Watching Cyan interact with his family in the Wrexsoul. You see how much he really did love his wife and kid. They even come back from beyond the grave to help him move on... this is on top of how he masqueraded as an enemy soldier for at least a year to keep a girl's heart from breaking.
Goes straight to heartwarmingly-awesome immediately afterwards: with the loving spirits of his family at his side in the form of the Masamune and his own spirit clear of all doubt, Cyan instantly unlocks all of his Bushidos. One of which instantly slays any and all enemies. At this point he is, as Japanese culture put it, ikkitousen: one swordsman to match a thousand, or, put more simply, a warrior without equal.
Strago and Relm reuniting is equal parts heartwarming and funny.
In the minor sidequest where you take Gau to meet his father(who is clearly insane), his father can't remember Gau at all, but mentions a dream about a demon baby he had once, Sabin, realizing the "demon baby" is Gau, gets naturally mad at tries to attack Gau's father... only to have Gau standing in front of his father, it didn't matter to him if his father didn't remember him at all, he was just happy to know he was alive...
"Fa-ther alive... Gau happy..."
Even sweeter is the way the whole party, even the normally-aloof Setzer, pitches in to try and make Gau presentable.