At the end of the prologue, Roxas finds out that he's actually the Nobody of another person — Sora. Depending on your interpretation, either he reluctantly agrees to merge with Sora to restore Sora's memories and self, essentially sacrificing himself, or simply has no choice in the matter.
Roxas:I guess my summer vacation... is over.
Sora begging to Sa´x on his hands and knees to see Kairi. Even more powerful with Japanese cultural connotation, as he is doing a bow known as Dogeza (土下座), which is reserved for when you've really fucked up, because by doing that act you literally saying you are worth nothing and purposefully and willingly putting shame and humiliation on yourself. Ain't Too Proud to Begindeed.
Goofy apparently dying in Hollow Bastion. Sure, he gets better like one cutscene later, but the loss comes completely out of nowhere. The way Mickey says "Goofy...?" is what really sells the scene.
Thanks to Final Mix, the Roxas boss fight. The true kicker is that a remix of Roxas's theme (titled "The Other Promise") is playing in the background, making the fight a really better sweet one. Watch it here.
"Part of Your World" can bring a few tears to an eye, despite the KH version being inferior to the film version, and the fact that the world does not get that much love. But anyone who grew up watching The Little Mermaid would enjoy being able to perform the song again, especially since it's Ariel's signature song.
The aftermath of the MCP battle, which is also a Heartwarming moment. After giving everyone a goodbye hug as a promise of meeting again, Tron just jumps into the pit where the MCP once stood, much to the horror of his friends. Sure he turns out okay a few minutes later, the scene is still very heartbreaking. The release of TRON: Legacy five years later, and then Kingdom Hearts 3D: Dream Drop Distance, over a year after that, made the scene even more unbearable to watch.
The memories re-uploading into Sora's head during the prologue. All the most powerful parts of the first game, clipped together with a slow piano melody.
Axel's death scene. Made even worse in the original unedited Japanese version, where he functionally explodes himself in order to wipe out the Nobodies trying to attack Sora.
Even more saddening when you realize how exhausted he is when helping Sora. Notice the sluggish movements, panting and slouching as he forces himself to fight. Talk about loyalty. He actually gives his life to protect the complete being of his best friend just because he cared that much. Also the fact that the way he's trying to play off his death is pretty much like him saying, "Hey, shit happens."
It gets better, or worse, depending on how you view these things. Watch the original, unedited scene; there are still flames flickering all over him as he fades. And the last little spark, as he fades away completely? It's right where his heart would be.
Since both Roxas and Xion are a part of Sora, they were forced to watch their best friend sacrifice himself, and both of them couldn't do anything about it. The reason you see Roxas breaking off from Sora afterwards is because he was enraged and saddened at Axel's death. The following scene becomes even more sad if you're using Bond of Flame while Roxas attacks...
And the sorrowful death scenes kept on coming. Saix dragged his half-dead body to Kingdom Hearts and begged for his heart. The kicker? He had a heart the whole time. And Ansem, yes, the real Ansem's death scene was also very heartbreaking, particularly due to his monologue of regret that came before it.
Roxas suddenly learning that he's the Tomato in the Mirror. "Sora... You're lucky. Looks like my summer vacation ...is over."
Now let's look back from here, Roxas spent 358 days before arriving at the fake Twilight Town, starting from when he was "born" to when he actually got there. Then he spent 7 days in the fake Twilight Town with the friends he never had, making it 365 days. 365 days = 1 year. Considering Roxas being sucked back into Sora on day 7, Roxas "died" on his birthday.
The manga adds one more heartbreaking line: "It was real to me..."
Roxas crying, through Sora, over Hayner, Pence, and Olette, the friends he never actually had.
And because it wasn't enough of a Tear Jerker the first time around in Final Fantasy X, the ending credits has to go and show Auron watching the residents of Olympus Colosseum celebrate and dissipating, unnoticed, back into pyreflies.
When Auron says "Perhaps... It's time I make my own story." Considering his role in FFX in regards to two generations of heroes and Auron's fate both times...
The Proof of Existence room in the Castle That Never Was. It's pretty much a Graveyard for the Organization, the portals to their personal rooms looking like tombstones and marked with symbols of their weapons. It's a room they need because Nobodies don't leave corpses behind — they need that room to mark to others that they did exist, even if they're no longer in existence.
It goes a little deeper than that. Consider this for a second: How do we, in this world, prove the existence of someone who no longer exists? We raise a grave for them.
The battles with Saix and Luxord are in their personal rooms, which as mentioned you get to via the Proof of Existence. You're essentially challenging and killing them in their own graves.
Not to mention all the cracked and broken markers to show how many members have fallen by that point, and how few are left.
And you realize after 358/2 Days: who's the one member who has no Proof of Existence at all? The one member who was never considered a member and is forgotten barely a day after she died? The series really drives home the idea that in the eyes of the Organization, Xion never really existed in the first place.
The prologue in its entirety, on second and subsequent playthroughs, especially after playing 358/2 Days. Knowing the real reason Axel is trying to stop Roxas, and what's going to happen to both of them... pass the Kleenex.
NaminÚ's parting words to Roxas - "Roxas! We will meet again. And then we can talk about everything. I may not know it's you, and you may not know it's me. But we will meet again. Someday soon. I promise!" And at the very end of the game, they do, smiling at each other through Sora and Kairi.
The ending often works tears, seeing the three friends finally brought home after all those years of separation.
Sora: W-we're back. Kairi:[holding out her hand] You're home.
Especially when one takes into account the original Japanese dialogue of that scene, which is literally translated exactly as above, but is culturally powerful.
Watching Mufasa die AGAIN, only this time in full 3-D. Thanks, Disney, it's not like that wasn't traumatizing enough the first time around.
Final Mix+, go watch the dream meeting of Roxas and Axel (warning: HEAVY spoilers). See them say "See you later". Cry when you realize they're just trying to cheer each other up, since Axel is going to disappear into absolute nothingness, and Roxas will be subsumed by Sora's personality.
Not helping matters is when you see the scene in a better quality, you'll notice Axel is crying.
When Sora finds Riku, and the way he grabs his hand and just breaks down and all he can say is "I FOUND you..." To see Sora, who's normally such an optimistic character, just break down can really get to you.
Gets even worse when you think for a minute and realize why he reacts like that: The last time he saw Riku, he was being locked in the realm of darkness and trapped there. The players know that Riku's escaped the realm and is all right, from playing through his story in Chain of Memories, but Sora sure didn't. Up until that point, for all Sora knew, his best friend could very well be dead, and he is relieved to see him alive and safe.
Luxord's death and his last line "How COULD you, Roxas?!" Made more upsetting after playing 358/2 Days, when he seemed to actually somewhat like Roxas, so it's obvious he feels very, very betrayed.
When Sora first goes back to the 100 Acre Wood, everything's all fine and dandy at first, then the Heartless come and poor Pooh Bear forgets Piglet.
Also when Sora enters the book for the first time and is forcibly ejected out before saying hello. Then discovering that the book has been torn again.
The last scene on Starry Hill. Pooh turns to Sora and admits he's been "trying to think of a way to make you not leave." That in and of itself is a sad line, but what made the waterworks flow was Pooh's next line:
Pooh: Sora, if I go looking for you, would I find you?
The end of the last Halloween Town episode, when it starts to snow and Sora sees himself and Kairi dancing.
There's a subtle one in Hollow Bastion. Right after the awesomeness that is The Battle of1000 Heartless, Sora leans against a wall to catch his breath. Just seeing this generally upbeat kid so exhausted (let's face it, you never even see him winded even after crazy boss battles before).
Goofy being hit with debris from an explosion. The reason Mickey's reaction is such a Moment Of Awesome is because many people pretty much wanted to say the same thing. And they would pay for this...
In Final Mix, there's a scene where Sora, Donald, and Goofy fight against Terra's Lingering Will. The fact that it's still there after all these years, waiting, and the only thing it seems to have left is its rage and hatred towards Xehanort is heartbreaking after playing Birth by Sleep.