Although it being retconned to hell and back in the sequels sort of ruined it, the final scene where the newspaper showed the Agent K was reunited with his lost love and Agent J's simple smile at it.
YMMV, but in an odd way, the memory J thinks up for Edger's wife after she's been neuralyzed is kind of heartwarming. K makes up that Edger ran off, and she'll decide she's better off, then J adds that she threw Edger out because he never appreciated her anyway and now she's going to make a fresh start. Considering what little we see, it seemed like an abusive relationahip, and the altered memory that she took charge and ended may end up empowering her in the future.
Men in Black II
After J questions Laura he takes out the neuralizer and tells her that it will wipe her memory. He then accidentally lets slip how lonely MIB life is. Instead of being afraid or alarmed, Laura shows sympathy towards J. J has a moment of weakness and claims that he'll wait until after he's finished his investigation.
Later when K is back and the pair go to her pizzeria, Laura thanks J for sending some MIB agents to guard her.
Towards the end, Kay thanks Jay for bringing him back to the MIB, saying "I missed this city." It's particularly heartwarming considering the movie was made in the aftermath of the September 11th attacks.
Men in Black III
In the third movie, the Colonel's "He showed me how important you two are" becomes heartwarming once you learn that he's Jay's father. Also, since Griffin likely showed him his own death, that makes everything he does one big Heroic Sacrifice.
"Your dad was a hero."
J being genuinely overjoyed when he finds K's past self, alive and well.
The last scene in Part 3 with K and J at the Diner, where K tell J it's been a privilege working with him.
And before that, the scene with Past K on the elevator up to the rocket. "I can see why I recruited you. You're a good man."
Seeing K singing The Empire State of Mind happily, which shows that that old curmudgeon finally has a genuine happy moment in his life.
For this troper, when O shows Jay, Kay and Griffin the jetpacks so they can reach Cape Canaveral as quickly as possible, Griffin gleefully runs up to them while laughing like a child seeing a new toy.
The animated series has a Christmas Special in which Agent K receives the first edition of Charles Dickens' A Christmas Carol. He cracks a smile, and looks like he's about to tear up for the first time in ages.
At the end of the series, J says "I'd make that Medal of Honor look good" after MIB works with the President to save the world from the Ixions. K answers "I know you would."
"The Sonic Boom Syndrome" has a Cold Opening in which J and K are running through a dark sewer, apparently chasing some beastie. Turns out they were the ringbearers for an alien wedding. Awwwwwww.
The one time K actually calls J by his name is very heartwarming. Especially J's reaction.
The episode "The Head-Trip Syndrome" has J mess up using alien technology to become super smart, but his brain will blow up. K and MIB try to find a way to save him, when it turns out someone is using time travel to erase the MIB. K suggests with them gone J will be healed and back in the NYPD. J only needs a second to think about it before saying that's not an option.