Ariel baring her soul to her friends in "Part Of Your World" and deciding not to lose hope about her dreams in "Part Of Your World (Reprise)".
Eric's reaction to saving Grimm during the shipwreck scene, which indicates that he's very fond of him. His terrified reaction when he realizes that Max is still trapped on board, showing how much he loves him. And the crew's horrified reaction when the boat blows up with Eric still on board, showing that they like and respect him very much.
Ariel saving Eric. Love at First Sight aside, she clearly can't bear to let an innocent person drown when she's able to help.
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.
There's a brief shot of Ariel's sisters, who are waving and cheering. It shows that they have no Fantastic Racism towards Eric and are happy for their youngest sister to have found love.
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 gotta 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.
Yes, Grimsby is the reason they're together.
A weird example, but Ursula's general treatment of her eel henchmen—she genuinely seems to love them as if they were her own sons, and they in turn treat her with loyalty and respect, carrying out her evil deeds with gusto. Indeed, what kickstarts Ursula's final showdown against the heroes is when Ariel's quick thinking leads to Ursula accidentally killing the two slimy little buggers.
Ursula: BABIES! (mournfully cradles Flotsam and Jetsam's remains in her hands) My poor little poopsies!
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...
A small one but Max recognizing Ariel's scent and the excitement he shows certainly counts, as does the big grin on Ariel's face when he licks her.
Flounder participating in creating the water fountain-like jets near the end of "Kiss the Girl" is rather touching to watch.
Ariel seeing her feet for the first time. Coming directly after the heartbreaking destruction of her grotto, her creepy meeting with Ursula, losing her voice, a horrifying near drowning experience after being transformed, and what must have been an exhausting swim all the way to Eric's castle, the look on Ariel's face when she wiggles her toes is enough to make your heart melt. Ariel always wanted to be a human instead of a mermaid, and after several of the most unpleasant moments of her life, her dream finally came true.
"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.
Ariel and Arista bonding in "Beached" while protecting the kingdom from evil alligators.
In particular, Arista revealing that she's always wanted to go on one of Ariel's adventures, despite having earlier dismissed them as "silly". It shows that despite often bickering with her, Arista loves for her younger sister and genuinely wants to spend more time with her.
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.
Master animator Glen Keane, who was the lead artist for Ariel, once told a story that he wasn't really proud of having animated her reaching up to the surface towards the camera on the last chorus of "Part Of Your World," feeling it was a little too cheesy, but he left it in to meet the deadline. Ten years later, a young woman told him that, as a little girl, she'd always wanted to reach forward, grab Ariel's arm and pull her out of the water during that scene. He's been proud of it ever since.