Sebastian's line during Ariel's first night at Prince Eric's place:
Sebastian: You are hopeless, child. You know that? Completely hopeless.
The wedding scene at the end, where everyone's happy and Ariel is saying goodbye to her dad.
Ariel: I love you, Daddy.
The moment where Eric bows to Triton and he nods back to him in approval.
And when Eric takes off in a rowboat with a harpoon to rescue Ariel from Ursula, starting off his Moment of Awesome:
Eric: I lost her once, I'm NOT going to lose her again!
Later on in the scene:
Ariel: Eric, you've got to get away from here! Eric: No, I won't leave you!
Grimsby advising Eric on his feelings for Ariel. He may be a Stiff Upper Lip, but he turns to the prince and gives him wonderfully heartwarming advice on love. And it would have worked too, if not for Ursula's magic.
Grimsby: If I may say so Eric, far better than any dream girl, is one warm, and caring... and right before your eyes.
It also shows how much Triton regrets how he's treated Ariel and never truly been able to understand her, despite his best efforts.
Eric teaching Ariel to dance to express her feelings in the stage version's number "One Step Closer". Highlighted even further when Ariel tentatively dances for him at the ceremony where he's meant to choose a bride. He has a Love Epiphany, claiming to have understood every word she said in her dance, then declares that he loves Ariel, even though he didn't see it before, and proposes to her on the spot.
Eric has a ton of these in the musical; his main song, "Her Voice" is a beautiful love ballad wherein he states his determination to find the girl with the beautiful voice who saved his life. And all he knows is that she has a beautiful voice.
Eric: And her voice Is sweet as angels sighing And her voice As warm as summer skies And that sound It haunts my dreams and spins me 'round Until it seems I'm flying... Her voice...
From the second film, when Ariel and Flounder are reunited after not seeing each other for years.
The kids that think her weird join her in the water.
From the third film: Triton and Ariel waltz happily; for a moment, Athena's back... then the entirety of his family an merfolk come to join the conga line.
"In Harmony" from "The Evil Manta" is one of the most heartwarming things ever.
When the Manta is trying to brainwash Flounder into hating Ariel, Flounder knows what he's doing but pretends to go along with it. The Manta leaves, Ariel catches up to Flounder and is all worried that he's no longer her friend, only for Flounder to drop the act and reassure her that he didn't buy the Manta's words for a second. Then this leads to a flashback of how Ariel and Flounder first met and became friends.
There is a story that when the film came out, Ron Clements received a letter from a New Jersey State Trooper saying that, after seeing the movie and being driven to tears by it, called his daughter to reconcile their own relationship, just as Ariel and Triton did.