There are quite a few Heartwarming moments in the film series, but the one that crowns them all is the original theatrical ending of Return of the Jedi. Of particular note is when Leia runs up and hugs Luke. For some reason this moment was shortened in the 1997 release onwards.
The Special Edition ending has the entire galaxy celebrating when the second Death Star is destroyed, which means that the Imperial army is gone and freedom and democracy have returned. It's just... the fireworks, the cheering, the energy... *sniff*
Made even better when Luke sees his redeemed father's spirit, returned to the man he once was before he fell to the dark side, or as the man he would have been had he not turned, depending on your preference of editions, sharing a friendly look with his old master, and then favoring his son with a loving smile.
The hug between Luke and Wedge, especially when you consider how Wedge and Rogue Squadron fought nonstop while Luke trained in the Force. Their relationship may have been mainly professional, but their embrace is warm and brotherly, each sharing their relief and thanking the other for their tenacity.
The hug between Leia, Luke, and Han at the end of A New Hope after the destruction of the Death Star is another great moment. Their exhilaration is just beautiful.
Also, Han coming back at the last minute to cover Luke so Luke can destroy the Death Star is a great Big Damn Heroes moment. "Great shot, kid, that was one in a million!"
Add to that after their intended second droid breaks down before even getting home, C-3PO quickly pitches R2 to Luke and Owen, ensuring his companion gets brought along with him (before of course returning to normal and snarking to R2 he better be grateful for it).
Plus, after spending the whole movie belittling him, C-3PO warns R2 to be careful as he heads out to attack the Death Star. "You wouldn't want my life to get boring, would you?"
And after the battle, upon seeing him badly damaged.
Threepio: (to Rebel technician) You must repair him! (to Luke) Sir, if any of my circuits or gears will help, I'll gladly donate them!
Echoed in Empire when they are parting ways on Hoth. Threepio begins by worriedly telling Artoo to take care of Luke before more quietly adding "And... do take good care of yourself."
Then there is the scene right after Obi-Wan saves Luke and the droids from the Tuskin Raiders. Luke and Kenobi find C-3P0 with a severed arm and the droid suggests he be abandoned while the others get away. Luke's reaction is simply that of course they are not going leave their companion behind. If that does not mark Luke as a hero you can respect, nothing will in the series.
When Han and Leia visit Luke in the sickbay after being attacked by the wampa.
Han: Don't forget, that's two you owe me, junior.
And it's brought up again in ROTJ:
Han: Hey, Luke, thanks. Thanks for coming after me. Now I owe you one.
It's blink-and-you-miss-it, but in the second movie, Vader says, "Only your hatred can destroy me." Think about it - there's a double meaning: it would kinda suck if your own son hated you, wouldn't it?
Luke's Shut Up, Hannibal! to The Emperor, but one line always stands out because of its multitude of heartwarming implications;
Luke: I am a Jedi, like my father before me.
Darth Vader's son, who has every reason to deny even the remotest connection with him, has just openly, and proudly acknowledged the relationship between them. Considering he's been trying to get his son on his side for two movies now, that's gotta mean something.
That the man he was, someone who has made so many terrible mistakes, is a person his son has decided is worthy of his respect and admiration. In fact, the implication of that statement is that he still considers his father a Jedi, and is therefore capable of redemption.
His son has utterly surpassed him; he's mastered himself and the ways of the Jedi, has gained power greater than his own without falling to the dark side, and has shown up alone to face the most powerful evil in the galaxy and has basically just won by refusing to give the Emperor what he wants- how proud would that make anyone feel, to see their son be so epically awesome?
Also involving Obi-Wan and Anakin, Obi-Wan's interior monologue in the novelization of Revenge of the Sith where, after mutilating Anakin he realizes that he can honor the memory of the man his friend used to be by protecting Anakin's kids.
“Anakin survived it,” Obi-Wan said. “Luke can, too. And I can — well, I could take him there, and watch over him. Protect him from the worst of the planet’s dangers, until he can learn to protect himself.”
“Like a father you wish to be, young Obi-Wan?”
“More an . . . eccentric old uncle, I think. It is a part I can play very well. To keep watch over Anakin’s son—“ Obi-Wan sighed, finally allowing his face to register a suggestion of his old, gentle smile. “I can’t imagine a better way to spend the rest of my life.”
The dark is generous, and it is patient, and it always wins - but in the heart of its strength lies weakness: one lone candle is enough to hold it back.