The Turtles' reunion with Splinter at the end of the first movie.
Just before the fade to the credits, while Leo, Donny and Mikey are dancing in celebration, Raph, who throughout the film was the most visibly shaken by Splinter's abduction, can be seen embracing his father again.
Not to mention Leo keeping constant vigil with Raphael the whole time he's unconscious and the "Kodak Moment" when Raph finally wakes up. Also qualifies as a Funny Moment.
Ralph returns it when he cuts off Leo's apology in a heartbeat. "Leo...don't."
Splinter: I am proud of you, my sons. Tonight you have learned the final and greatest truth of the Ninja: that ultimate mastering comes not from the body, but from the mind. Together, there is nothing your four minds cannot accomplish. Help each other, draw upon one another, and always remember the power that binds you. The same as what brought me here tonight, that which I gladly return with my final words: I love you all, my sons.
Splinter later helps Danny redeem himself by telling him that "...all fathers care for their sons."
The typical lecture that Raphael expects from Splinter instead becomes a very moving scene in the first movie. The somber music and Raph's speechlessness help to double this moment as a bit of a Tear Jerker.
Splinter: My master Yoshi's first rule was... "Possess the right thinking." Only then can one receive the gifts of strength, knowledge, and peace. I have tried to channel your anger, Raphael, but more remains. Anger clouds the mind. Turned inward, it is an unconquerable enemy. You are unique among your brothers, for you choose to face this enemy alone. But as you face it, do not forget them. And do not forget me. I am here, my son.
Danny and Charles' reunion. Also falls under Heartwarming.
Danny: Dad, I'm fine. And Dad, it's just Dan now, okay?
(Father and son share a tearful hug.)
Leo & Raph's reconciliation in TMNT
Raph hands Leo a pair of katanas to replace the set which were broken in their fight Raph: You're gonna need these if you're gonna lead us out of here. Leo: I'm gonna need you, too.
In the same movie, Leo coming back from his pilgrimage and being welcomed by Splinter. Leo and Splinter calling each other Father and Son, respectively, can make any fan of the series shed happy tears.
Turtles Forever when all three turtle groups congratulate each other at the end of the film.
Leo: Bros? Hey! We're bros!
Casey Jones in the first movie. First off he barely knows Raphael save for one fight with him, however after spotting him in danger he jumps into action. He didn't even meet the other three but stalled the Foot long enough for them to get to a safe distance. And that is before he was established as their friend.
His own mini-arc in the film counts. At the beginning, he is a bitter vigilante who violently punishes all criminals regardless of who they are. When he finds the Foot Hideout at the climax and finds out that most of them are young kids, he convinces them to abandon the dysfunctional family of dishonour that is the Foot Clan.
In the second movie, while the mess the turtles make in April's apartment is Played for Laughs, they work together to clean the place up for her, showing that they care about her.
Splinter in all incarnations is heartwarming personified. When he's not being stern or fighting, he is often a parental figure. Not just to the Turtles or April, but to anyone that he feels needs it. A big example is in the first movie, when he talks Danny into resigning from the Foot Clan, despite the fact that Danny is responsible for burning down April's home and nearly getting the Turtles and April killed.
This is especially shown when in Turtles Forever, the 2003 Splinter scolds his turtles for making fun of the original cartoon turtles, saying that they are all his sons. And it gets better- the original cartoon Splinter says the same thing to 2003 Leo and Raph. No matter where the Turtles are, they can always find a father figure in Master Splinter.
Meta: the making of the fan documentary Turtle Power helped Kevin Eastman and Peter Laird renew their friendship after it had cooled, and the epilogue shows them together for the first time in many years.