Okay okay I might as well put this in because we're all thinking it. The Ship Tease between Ash and Misty is sooooo obvious. Melody has been ragging Misty for the entire movie about Ash being Misty's boyfriend. Finally Misty decides it's time to go out into the apocalyptic storm to rescue her man. "Ash is never really alone, because he has me." Gets me all misty-eyed (pun intended) every single time.
It doesn't even need the Shippy elements to work, because even without them you can tell these two are True Companions. Before the Original Series had ended, Misty was the human friend who stuck by Ash since Day One (up until their long journey together was finally over), and travelled with him through more regions than her "successors". It's small wonder the audience is attached to the dynamic of the original travelling group. Say what you will about shipping or whatever, this scene defined the relationship between Ash and Misty.
Misty and Melody become "official" friends at that point too, which is doubly heartwarming.
Ash and his mother near the end of the movie. Delia scolds him for taking such a risk with his life, then explains that while he was saving the world, she came so close to losing her whole world. The whole exchange is just a big ball of heartwarming.
Delia: And just remember. Every day...you're my hero.
The little conversation among Jessie, James, and Meowth as they plummet to what they think is their death after jumping off of Lugia to allow it to fly faster.
(All three move from holding hands into a group hug)
Jessie: Let's not say goodbye.
James: Let's just say...
Meowth: ...we're gonna die.
Lugia's farewell line: "The Beast of the Sea had been tamed. The fate of the world could not have been in better hands."
A minor one near the start: when the boat hits the rough weather, Tracey's warrior Scyther reaches out to steady Venonat. Seems simple, but Scyther is both the newest addition to the team and a notoriously violent species.