Cyan's storyline. The death of his wife and kid as part of the poisoning of Doma Castle wasn't too bad, emotionally. It's soon followed, however, by a level where you accidentally board a train to the hereafter. You fight your way to the engine, defeat the train, and it agrees to take you back to the land of the living. But while it drops you off, it picks up all of the residents of Doma. Cyan gets one last desperate goodbye to his wife and son, the last words you hear from them as the train rushes off into eternity is the son promising to look after his mom. Just to make sure you're feeling Cyan's pain, the game lingers on the scene for a good 30 seconds afterward while Cyan stands at the edge of the depot in despair, staring at where the train once stood.
Celes' attempted suicide if Cid dies and the cutscene after getting the Phoenix magicite are particularly powerful. The scene described in your spoiler sets up a Heartwarming Moment when Celes survives and discovers Locke's bandanna, setting off the rest of the game's chain of events.
Rachel: Please, let go of the chains that bind your heart... I release you... Give your love to to the one who now dwells within your heart... Love her... as you loved me...
If you fail to get Locke back in the World of Ruin, Celes' ending is the same as the heartwarming ending, except that Setzer saves her and yells at her for going back for Locke's bandanna. After he leaves, she looks down (at it) and says, "Locke, promise me some day you'll look after me again." before walking out the room.
When Terra talks to Banon in the Returner Hideout, he asks her if she will be their "last ray of hope". If you say no three times, he says "I thought as much" and Terra walks in and says "How can anyone look to me as hope".
Edgar and Sabin's backstory, with their father dying. Sabin thinks no one cares about their father and wants his freedom. He urges Edgar to run away from the kingdom with him, where they can be free of the kingdom. Edgar wants his freedom just as much as Sabin, but can't bring himself to leave the kingdom without a leader. So, he suggests a single coin toss. Whoever wins will choose his own path, with no regrets. He rigs the coin toss to give Sabin his freedom, choosing to shoulder the burden neither of them wanted. Now, that is brotherly love!
General Leo's death scene.
Terra's words at his funeral.
"General Leo... People only seem to want power, don't they? Do they really want to be like me? I... I wanted you to teach me so much more..."
The entire interlude into the World of Ruin is unbelievably depressing. Celes wakes up on an island after being in a coma for a year with Cid taking care of her. Being the closest thing she has left to a family, she begins referring to him as her grandfather. But it turns out that Cid is ill from having lived on nothing but rotten fish for the past year. If that wasn't bad enough, try leaving the area and visiting the world map. The island is absolutely desolate, and the music only further drives home the feeling of isolation and despair. On top of that, any monsters you encounter won't attack so much as pathetically try to cast spells they don't have enough MP for and then drop dead on their own before you can probably attack. And if that isn't bad enough, let Cid die. He doesn't give any last words; Celes simply finds him silent and unmoving. She doesn't even realize he's dead at first. Now completely alone and with no hope of ever seeing civilization again, Celes climbs to the top of a cliff and jumps off. Damn.
And then there's the first time players who experienced Cid's death, and then learned it was literally all their fault, as it's possible to save him.
Immediately after leaving the desolate island, you arrive very close to Albrook. The man in the cafe there says that he thinks back on how good the world once was—leading to a brief flashback showing people dancing and relaxing in the cafe—and then realizes that it will never be the same way again.
Aria di Mezzo Carattere: Guaranteed to make you bawl if you take the lyrics of the song to heart, and have lost someone you loved.