Marle: I used to call you Daddy? King Guardia XXXIII: You did.
The Crono and Marle moment at Death Peak where Marle hugs Crono after he was brought back to life and tells him how much she misses him. Lucca, the Unlucky Childhood Friend, takes her place if you decide not to put Marle in the party.
Also, Lucca doing the same. Something about the way that Lucca yells at Crono for "being stupid" and proceeds to collapse on him ... is powerful.
If you bring Magus to the Surviving Village in 12,000 B.C., Alfador will run up to him and constantly meow. He still remembers his master Janus despite how much his master had changed.
Even better is what is shown in one of the alternate endings where both Janus and Magus (as the Prophet) interact, which shows Alfador genuinely confused at which 'Janus' to follow!
To be fair, it's been very little time to Alfador; but it's still a heartwarming Pet the Dog moment for Magus, because it shows that he hasn't changed so much that his cat wouldn't recognize him.
The mayor of Porre starts the game as a greedy, cruel man whose own children hate him. Then you teach his ancestors the value of generosity, and return to the present to find that he's now a perfectly nice guy whose kids say "My dad is my favorite person in the whole world!"
Saving Lucca's mother from the rogue machine. Sure, it's completely optional, but it still feels wonderful. Granted, that could be because the other option packs a hell of a Player Punch.
If you do fail, there's still a heartwarming moment when you get back to the present: Robo offers to let Lucca remove his legs and attach them to her mom and he can be stuck with tank treads. ... It's the thought that counts, right?
The whole aftermath of that mission is a Crowning Moment of Heartwarming, really. Robo volunteering to make Fiona's dream of replanting the forest reality; warping to the present and finding that it worked and the desert has changed into a big forest with a shrine dedicated to Fiona and Robo in the middle; reactivating Robo who happily greets you after spending 400 years without his friends; the group relaxing around the campfire, talking with each other... hell, even the normally silent Magus chips in. Then, after Lucca has saved her mother, you can go visit her. The woman who spent her days looking out the window and slowly drifted away from her husband and daughter has changed into someone who happily talks about the beautiful weather and how she thinks of making a walk before finishing the housework.
And then demands Taban dance with her at the Moonlight Parade.
The scene in 2300 AD where the party brings the seed back to the villagers so they can grow crops. When asked about why they are so optimistic, Marle responds that it's because they "are healthy/have hope". Then the villagers all begin to show optimism for the first time since the 'Day of Lavos' in 1999 AD, making this one of the most powerful scenes in the game. The fact that the music theme that plays at that moment, appropriately named "The Day the World Revived", doesn't hurt either.
If Frog is in the party when you meet with Magus after Crono dies, he will spit out some invective about Crono's foolishness, prompting Frog to draw his weapon. A tense standoff ensues, and you can let Frog fall victim to his hatred and grudges, triggering an awesome Duel Boss. Or, you can back off, and Frog will sheathe his blade, letting go of his hatred and desire for vengeance, telling Magus that violence will not bring Crono back. It's a wonderful bit of character development on his part.
Robo is badly in need of repair after his brother robots all turn on him for being a "defect" and brutally beat him in front of Crono and Lucca. Offhandedly, Lucca asks him during the repairs what he wants to do in the future. Robo's response?
Robo: "No one's ever asked me that before!"
The ending that depends on you reviving Crono, crashing the Epoch inside Lavos, and defeating Lavos fully. Crono and Marle are carried away by balloons and they float through the night-time sky together.
Who would want to visit Manolia Cathedral right after Crono died? You know, that place in 600 A.D. that you cleared out in the beginning of the game after which it stood vacant and became a location of little to no interest. If you do happen to visit it after the aforementioned incident, you will meet a nun who has taken up residence there. Talk to her and she gives you a few words of encouragement, making it a completely random but very powerful moment in the game.
Nun: In this world, there are no rains that fail to lift - no nights that are not dispelled by dawn. What's important is to trust in those you love, and never to abandon hope. The person you cherish is no doubt watching over you from some distant place.
During the ending, when everyone's going their separate ways, it's mentioned that since Robo came from a desolate future (which the actions of our heroes have averted), he may not exist in the new timeline. It's a genuine Tear Jerker, with a quietly mournful lullaby playing as Robo comforts a distraught Lucca before stepping through the Time Gate, and possibly erasing himself from existence. However, if you get the ending where Crono, Marle, and Lucca grab the Epoch to look for Crono's mother and his cat, one of the time periods they soar through shows Robo sitting with his companion Atropos on a verdant hillside. Looks like the future did have a place for him, after all!
Frog's final words with Cyrus; Cyrus dispels all doubt within his friend, reassuring Frog about the latter's skill as a swordsman, and as a friend. Frog clearly could not spit out what he wanted to say, but he and Cyrus were so close that Cyrus seemed to already knew what Frog wanted to say before departing for good. This is even more heartwarming if you tried visiting the ruins in the present; Cyrus needed the closure himself and 400 years of his own worries and self doubt turned him into a vengeful avatar of hate who's only lucid moment was when he realized he knew someone named Glenn. In essence, the scene wasn't for Frog or Cyrus, but both. Masa and Mune of course lampshades the hell out of this.
Think about what Lucca is doing when she goes back in time to prevent her mother's accident: Originally it was the accident and the crippling of her mother that turned her mind to science. By going back to that moment and preventing the accident, there was no guarantee that she would have ended up with the same interest in science. By doing that and becoming a completely different person, what's to say that the many important events in the game that rely on her would have happened? She basically chose her mother over the entire timeline!