Anything that shows Shovel Knight and Shield Knight together is either adorable or tear jerk-ish.
An easy to forget one goes to the Order of No Quarter near the ending: Being defeated by Shovel Knight in succession, they are left holding on to each other in a Chain of People; they hold each other, especially members who are falling after being defeated, and don't let go no matter what.
And then Shovel Knight is able to help them up onto the ledge even though they were all trying to kill him mere seconds ago, and with no guarantee they wouldn't start attacking again. It's completely optional, but undoubtedly the shovelrous thing to do.
After the first phase of the Final Boss, Shovel Knight can hold Shield Knight, who is falling down to the ground just like in his dreams. Shield Knight's reaction cements the heartwarming feeling.
Shield Knight: ...You caught me. Shovel Knight: If only I had known, I would have come sooner! Shield Knight: You knew I was still there, even though it seemed hopeless, you never gave up on me! Shovel Knight: I will NEVER give up on you. I will follow you to the end of the world. *The Tower of Fate shakes violently* Shield Knight: It appears that maybe you have.
There's something about seeing all of the Wandering Travelers ready to fight together in the credits sequence that's just heartwarming. Especially The Baz, seeing as he finally got what he wanted: an order of his own to join.
The final battle, in which the player has to work together with Shield Knight to defeat the boss. The only way to reach the boss's weak point is by bouncing off her shield, and one screen-filling attack can only be evaded by hiding behind Shield Knight. It's heartwarming getting to see in gameplay how these two work together.
Special mention to the ending. Where Shield Knight is revealed to have survived, limps all the way to Shovel Knight's camp from presumably the Tower of Fate, sees that he is resting, and just lays down to rest beside him.
There's a good dose of Heartwarming in Hindsight in the ending after beating Specter Of Torment. Specter Knight, having had everything taken from him by The Enchantress in the course of his campaign, is finally free to live, er, un-live his undeath.
Plague of Shadows
Damn near everything involving Plague Knight and Mona.
After defeating King and Specter Knight, Plague Knight returns to his hideout and finds Mona dancing alone... His awkward reactions and answers when Mona tells she has no one to dance with sells the whole thing. The heartwarming goes Up to Eleven in the ending, showing Plague Knight waltzing with Mona using the skills he leaned from the Troupple King.
A simultaneously heartwarming and cute moment: if you hold the down button anywhere, Plague Knight starts dancing in place. Try it in Mona's room, and he's too embarrassed to do so, instead drumming his fingers together and looking obviously shy. It's cute.
After defeating Shovel, Mole, and Treasure Knight, Mona clearly pretends that the lever to their machine is jammed, wanting Plague Knight to "help" her. Plague Knight goes over to her, placing his hands over hers (as noted by text)... and then the lights go out, followed by Percy being a Moment Killer.
The entire reason Plague Knight wanted to create the Ultimate Potion: To become strong enough so Mona would love him, unaware that she already did. Crosses into a bit of a Tear Jerker if the player encounters Black Knight on the map, which has Mona run off and not appear for the rest of the game. Plague Knight clearly misses her, not even happy (at first) when he collects all 8 essences. Comes to a head in the endgame where, after defeating the Corrupted Essence and regaining the Ultimate Potion, Plague Knight discards it, allowing its energy to blow up the Tower of Fate, claiming that since Mona loves him, he has everything he'd ever need.
A few ending scenes showing the newly hooked-up couple enjoying their relationship, including having a romantic dinner on the Flying Machine (with Propeller Knight as the waiter), and later on, coming together in the best way they know: SCIENCE!
After defeating The Baz, Plague Knight can hire him as one of his minions. Once hired, The Baz shows an incredible amount of loyalty. It really shows how much he just wants to belong.
Specter of Torment
The truth of the locket. It represents the promise Specter Knight made when he was alive. The promise to protect his partner's son, Reize. And he serves the Enchantress only to keep this promise.
And he also sacrifices most of the amulet's power just to set Reize free from the Enchantress when she uses her power to convert him. The same power that was protecting his free will. And then Specter knight picks up Reize and carries him in his arms across the land to a place where he will be safe, before finally, finally succumbing to the Enchantress's dominion.
The post-credits scene has Specter Knight having one last flashback as Donovan, when he received the locket from his partner, Luan. Here is how they sound off on the campaign, doubling as a bit of a Tear Jerker knowing what ends up happening:
Luan: This is an old family keepsake. You are family to me, and... I was hoping you would be family to Reize.
Donovan: Luan... I would lay down my life to save his. I promise.
During the ending, the Plague Knight section is called "Bubbling Chemistry". He's building his undertown base with Mona, pulling up parts of it with a chain, when she accidentally doesn't pull her part of the chain hard enough, and, well...
Also, during the credits, Polar Knight looks after a wounded Terrorpin.
The Red Skull Quest. The red skeleton, Red, is actually looking for his wife, and when you collect her unique red skull they actually get reunited and you're treated to a scene of them dancing together. And when you collect all 100, he gives you all the stuff you had as Donovan and wishes you luck against the Enchantress. They even appear in the Lich Yard ending in a house, being the only two undead that don't try to run out the villagers.
The Dark Acolyte that was helping you throughout leaves the Enchantress and becomes the Troupple Acolyte during the ending sequence.
During the ending, all the main NPCs flee the Tower of Fate upon seeing Specter Knight emerge with Reize in his arms, flying towards the exit. Not heartwarming on it's own until you remember that upon collecting all the red skulls, Red revealed that he could tell that Specter Knight was going to betray the Enchantress and told him to be careful. It's likely all the NPCs realized this and fled once they realized Specter Knight had failed to stop her, despite having nothing to fear from it themselves. Their loyalty as friends to Specter Knight was stronger than any loyalty they had to the Enchantress.
If one visits Specter Knight's room in the Lich Yard, they can see two hammocks, two chairs around one table, and spare articles of Luan's vest in the hidden closet. The implication that they lived together can really remind the player as to how close they were as partners.
You can play with the Memmec living in Specter Knight's headquarters. But proof Specter is Not So Above It All? Press down close enough to it and Specter will start petting it. After making sure no one is looking, of course!
Shovel Knight Showdown
The backstory for Showdown is that the NPCs from Specter of Torment try to help Specter Knight defeat the Enchantress because he's their friend.