The fact that Marlin let's go of the whale's tongue. He doesn't realise the whale is trying to help them, meaning Marlin actually attempts suicide!
Coral's death in the beginning.
This is one of those films with undertones that the parents will understand just a little bit better than the kids.
Nemo: It's okay, Dad, you can let go.
"It's okay, Daddy's here, Daddy's got you."
The moment when Nemo tells Marlin "I love you, daddy" and Marlin says "I love you too, son".
The completely heartwrenching moment when Marlin sees Nemo pretending to be dead in the plastic bag and, believing his son to really be dead, gives up and leaves in defeat. Dory's reaction as well:
Dory: Oh my goodness....
For that matter, when Nigel returns Marlin and Dory to the ocean. He was so eager to help Marlin reunite with his son, and then all he can do is apologize.
"I have to tell him how—old—sea—turtles—are!", although that one comes across as narm.
"I promised I'd never let anything happen to him."
The scene where Dory actively laments her forgetfulness and expresses fears of forgetting her new-found friends:
Dory: Please don't go away. Please? No one's ever stuck with me for so long before. And if you leave... I just, I remember things better with you! I do, look. P. Sherman, forty-two... forty-two... I remember it, I do! It's there, I know it is, because when I look at you... I can feel it. And-and I look at you, and I... and I'm home! Please... I don't want that to go away. I don't wanna forget.
Marlin: I'm sorry Dory. But I... do.
Made even worse by a scene later when Nemo finds her scared, alone, and desperately trying to remember what she needs to do.
The best testament to this film's ability to make you cry is that the first tearjerking moment comes before the opening credits, thanks to some amazing voice acting by Albert Brooks:
Marlin: I swear, I will never let anything happen to you... Nemo.
It gets worse when Marlin and Dory are in the whale's mouth, with Marlin finally realizing how overprotective and limiting he sounds. Some paraphrasing here.
Marlin: You can't do this! Dory: Yes, I can! Marlin: No you can't! You say you can do these things, but you can't, Nemo!
Any time the Nemo Egg theme starts playing, (the beginning right before the opening credits, when Marlin or Nemo contemplates how to solve the main conflict, the scene where Nemo first hears about his dad's daring adventure to rescue him, and the very end).
Thomas Newman, who also composed the soundtrack the Andrew Stanton-directed WALL-E, has a knack for establishing ethereal atmosphere that can also tug at the heartstrings. The music that plays during the scene inside the whale is considered by Stanton to be the best track in the movie.
Bruce lamenting that he never knew his father. Yes, it's Played for Laughs, but for viewers who grew up without either parent, it's surprising to empathize with a shark. Though, he is a great white, so his father probably would've eaten him, but still. A bit of Fridge Brilliance there, since real life sharks don't raise their young and abandon them at birth.
Although Marlin is the undisputed master of the Adult Fear trope, Gill has his moments. Watching the clear desperation on his face as he watched Nemo be scooped out of the tank was enough to rival any parent. (He calls him by his real name which he only does a few times throughout the whole movie when he promises him that he's not gonna go belly up.)
The line, "Dad?...I don't hate you."
Upon rewatching, the scene where Marlin and Coral name their eggs is quite tragic.
In the E.A.C, when Marlin first sees Dory, facedown (he thinks she's in a coma, but she's just playing hide and seek), and it's obvious that he's completely guilt-ridden over her injuries.
The scene where Marlin tells his story to a group of turtles in the E.A.C., which then spreads throughout the ocean.