The "Total Eclipse of the Heart" sequence in Episode 36 was genuinely pretty touching.
Yugi also claims that his friendship with Joey means more to him than card games. Given that card games are played to such obsessive levels in this show, that's saying a lot.
"Must... risk life... for best friend..."
The ending, when Joey and Yugi start Waxing Lyrical, leading into the song "No Matter What" playing, and Kaiba, the last person you would expect, says he loves this song.
Marik and Bakura's death scene in which they sing "Stand by Me".
In Season Zero, Episode 1, after spending the earlier part of the episode bullying Yugi, it's incredibly heartwarming to see Joey and Tristan rush to Yugi's rescue after he is beaten up by Ushio though the moment is weakened by The Reveal that Yugi wished for Ushio to kick their assess
Joey: It's ok, Yugi. We're here now. Your wish came true, man. Tristan and I are gonna protect you.
"YGOTAS Sails Away" too.
Especially when, in the midst of offering several different things in return for Bakura agreeing to sing with him, Marik finally says, "What if I told you I loved you?" Bakura is first shocked... but then, a little later, he does sing after all. The fact that his voice softensever-so-slightly when he reacts makes it.
The part where Mako Tsunami gets back together with the ocean, especially because the ocean finally speaks.
Also Joey's surprisingly legit relationship advice he gives to Mako.
Joey's speech to Johnson in Episode 50 after being accused of copyright infringement (also doubles as a Crowning Moment of Awesome):
Joey: Yeah, I have a closing statement. Maybe we have committed copyright infringement. But you gotta know we've done everything in our power to support the Yu-Gi-Oh franchise! And if it weren't for us, I don't think the show would be nearly as strong as it is right now!
Johnson: And where is your evidence of such?
Joey: Look around, Johnson! There are more Yu-Gi-Oh fans now than ever before! And the more you try to stifle our creativity, the more we'll try to express our love for a show that's about more than children's card games! It's about fighting for what you believe in, and I believe in this show and its fans now more than ever! Because they believe in me!
"It's okay, Serenity. I've lived my life as nothing but a side character, fit only for spouting brainless quips and catchphrases. But now, I get to die the way I've always wanted: falling, screaming like a girl, into a pit of boiling hot lava, flesh being seared from my bones. If I do survive, I'll probably be unrecognizable. But if I do die, it'd be as a main character. It's funny; I used to think my voice gave me super strength. But now, I know the truth. And it's my love for you is what really gives me super strength."
Also, Tristan's Heroic Sacrifice was to protect Duke, whom he hates not only for trying to seduce Serenity, but for being a witch. Yet when Johnson declares what Tristan thinks is his intent to attack Duke, Tristan sacrifices his Deck Master and loses the game to save him.
From Episode 53:
Noah: Admit it. You don't really care for him.
Kaiba: Yes, I do!
Still on the topic of Seto and Mokuba, in Episode 14:
Seto: I have to beat Yugi if I ever want to see my little brother again. I miss you Mokuba.
In a deleted scene of Melvin'd, Melvin calls Florence out on his hypocrisy in trying to prevent Melvin from destroying the world when Florence has the same goal. Instead of invoking The Only One Allowed to Destroy the World, Florence instinctively answers, managing to back himself in a corner, leading to this speech:
Episode 52: Hobson helping the Kaiba brothers kick Gozaburo out by giving his stock to Seto. Not only does it arguably make more sense than the canon (seriously, who would give stock to a 5-year-old?), but it's a genuinely clever Chekhov's Gun set up by flashbacks from episodes that were made months ago. Namely, that young!Mokuba was the only one who was kind to Hobson.
Joey getting annoyed at Yami for making jokes about Mai when she's in a coma. Am I right?
Episode 55 continuing to show us that for the all the crap and mean things he says to Mokuba, Seto really does care about his brother... even if he finds it hard to admit it in front of everyone.
Episode 56 ends with the gang mourning the end of 4Kids — note that, as the Hilarious in Hindsight entry goes into detail on, in real life over the course of Season 3, 4Kids had lost the rights to most of their dubbed shows including Yu-Gi-Oh! and declared bankruptcy. The cast's comments are quite clearly meant to be taken at a meta level.
Kaiba: Well, I guess that's the last we'll see of 4Kids. Yugi: It's weird. I'm actually kind of sad to see them go. Kaiba: Yeah, after all, they did provide us with hours of entertainment. So what if they censored a few things? They were just doing their job. Yugi: And without them, we wouldn't have Yu-Gi-Oh Abridged. So from now on, let's remember them as the guys who gave us fond childhood memories — and unfortunately, the One Piece theme song. Mokuba: Good night sweet dubbing company. And a flight of Winged Kuribohs sing thee to thy rest.
In Marik's Evil Council 4, Melvin manages to kill Yugi, Psychoshipping becomes canon, Bakura is voted Villain of the Year, and everyone agrees that Florence is the manliest name ever. Bakura says it's the happiest day of his life, and he actually sniffles. Sniffles. Bakura.
Marik's talk with Odion during the duel between Yami and Melvin. While it wasn't quite the heartfelt apology that his canon counterpart gave during the same scene, Marik still managed to swallow his It's All About Me attitude long enough to at least let Odion know what a great brother he was. The fact that Marik intended for these to be his Last Words makes it double as a Tear Jerker.
Meta, but this comic a tumblr user made for LittleKuriboh in response to the complaints about the Marik Plays series seeming to take more precedence over the main series after the move to Team Four Stars site: Marik is stalling for time so that the series won't reach its Foregone Conclusion, Bakura's death at the end of the series. He just doesn't want to lose his only friend.
Two of LK's abridged thiefshipping fics get stunningly WAFF-y.
In episode 62, Joey's breakdown when Mai won't wake up. It's a major development considering earlier he only cared about her breasts and all he had to say about her fate was "Well ain't that a kick in the teeth".
"No...Mai...This can't be happening. I love you. I was gonna spend the rest of my life with you..."
While it seems to only be Yami threatening Weevil, he seems to acknowledge that Yugi operates as his moral compass, keeping him from going off the deep end.
Episode 74 has this poignant speech from Yami during his and Yugi's duel, showing just how deeply he cares about Yugi beneath all the Awesome Ego and snark. It's also a rare case of him apologizing for his jerkass behaviour.
In episode 76, Rebecca mentions that Dungeon Dice Monsters is her favorite board game and uses its system as a basis to explain the card game to Duke. Duke is so touched he says he may love her.
In the fanfic ''Fun in Yugi's Mind'' by indecisivepancake we see Florence come to Bakura's defense when Melvin destroys his beloved "Change of Heart" card.
The processing video was obviously in-universe about Yami's loss in ep 71 but it is very clearly directed at viewers who are devastated by Donald Trump's unexpected victory and not for them to give up hope. Also, the author's followup video, which goes into more detail while touching on his own struggle with depression.