The entire montage of Jim's growth and the development of his friendship with Long John Silver, inter cut with flashbacks of his father abandoning him to show Silver as his father figure. Also set the lovely I'm Still Here by John Rzeznik.
The ending: Although these days it's practically required by Treasure Island remakes to turn Silver into a surrogate father to Jim Hawkins, this movie did it in a warm fuzzy believable way that gets you everytime.
"Now you listen here, James Hawkins. You've got the makings of greatness in ya. You got to take the helm and chart your own course. Stick to it, no matter the squalls! And when the time comes you get the chance to really test the cut of your sails, and show what you're made of! Well, I hope I'm there, catching some of the light coming off you that day..."
And he was.
Silver: (After Jim saves them all): DIDN'T I SAY THE LAD HAD GREATNESS IN 'IM?
And for the cherry on the icing on the cake, "Always Know Where You Are" playing over the credits.
Silver sacrificing the treasure in order to save Jim.
Two things in particular: At the end, Jim looking up at the sky and seeing a cloud-form Silver smiling down (complete with a star for his eye), and this quote:
Silver: Look at you! Glowing like a solar fire. You're something special, Jim. You're gonna rattle the stars, you are!
Doppler and Amelia's growing, if subtle, relationship throughout the movie:
The moment after escaping Treasure Planet's destruction via a portal back to the Spaceport, Doppler and Amelia hug in joy...then after a few seconds, they look at each other...and smile. D'awww.
A few scenes later we find out that they are Happily Married with four kids: Three girls who look like their mom and one boy who's the carbon copy of his father.
Amelia: "Doctor...you have wonderful eyes".
A little moment during the supernova scene, as they ride the last wave out of the dying star: Jim is pressed against the mast and his lifeline, scared to death, and Silver has both his cyborg arm and his whole body around Jim, protecting him from whatever goes flying after them. And Jim, the boy who plays with danger all alone, doesn't even realize he has someone protecting him until they're safe and he opens his eyes.