Every time Mario and Luigi share a hug when they meet after being separated.
Also deserving of mention is their hugging Princess Peach after she is kidnapped in Teehee Valley (which you can repeat, even multiple times, if you let her get captured again). There's no unique sprite, so it's staged as just the Mario-&-Luigi-hugging sprite over top of Peach's, but it gives the impression of a Group Hug, which is very touching as it really shows how much the plumbers and princess appreciate each other's company and help.
When either one of the brothers suffers a KO in battle, the one still standing gets an absolutely horrified expression, rushing over to the fallen one and supporting his unconscious body.
The premise of Minion Quest is that all of Bowser's Minions are out looking for him after he vanishes, and band together to rescue him. And they're all being banded together by a single Goomba, who becomes the captain, driven by his loyalty to Bowser. What's more hearwarming is the fact that despite being shitted on and trashed on their entire lives as being useless and the lowest of underlings, one of their kind manages to rally the entire army back together and rescue Bowser, and manages to live up to the title that his small group only bestowed upon him because no one else wanted to do it, and earn the respect and admiration of the ENTIRE ARMY. He became his own character, and he deserved to be promoted at the end, even if he ended up losing his position in the end. Not to discount anyone else, but Captain Goomba, you are the MAN.
Also the story's final image has a minor one. As Captain Goomba is fleeing from the incredibly furious Bowser, his fellow captains, some minions and even the Koopalings, can be seen cheering him off as he runs away.
After Captain Goomba is attacked by the amnesiac Bowser in Chucklehuck Woods, he falls down and cries for hours. When he gets up again, the other captains are still there, and cheer him up. It's a surprisingly adorable scene.