Oliver accepting the fact that his mother Alicia is never going to come back no matter what he does. After Shadar is defeated, she appears as an apparition before him to console him that there's nothing to fear before she disappears into the afterlife with Shadar.
Allie's, Oliver's mom, death. We all knew it was going to happen, but it happens so early that it can be unexpected. Even worse, there's a subsequent scene that shows a younger Oliver and Allie, with the latter saying they'll be together forever. Pass the tissues...
Basically, any flashbacks between Oliver and Allie makes her death that much more heart-wrenching.
The ending theme, Kokoro no Kakera ('Fragments of Hearts'), in particular the English translation: "If I could only be with you once more, and hold you tightly to my heart. We could walk this road together, and never ever be apart."
Seeing Gascon leave his brother in the past. It's subtle, but you can really tell how painful this is for both of them. By now, most players will have figured out he's Swaine, which aleviates this somewhat, as they know he'll make good on his promise to meet again. So the game pulls a double whammy: you all return to the palace... and find the Emperor dying. Swaine, i.e. Gascon's future self, now has to watch his own father die. And he knew Swaine was his son all along. You see the guy break down for pretty much the first time in the entire game. "I just wanted to make you proud..." Sniff.
Quite a few of the Tales of Wonder, found in the Wizard's Companion and unlocked as the game progresses, can be pretty heavy themselves. Case in point: "The Bear-Man and the Princess's Tears."