The 'Total Eclipse of the Heart' sequence was genuinely pretty touching.
Also Yugi claims in that episode (36) 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, saying that he loves the 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.
Tristan's speech in episode 51.
"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."
And then there's the adorable Star Wars reference:
Serenity: I love you.
Tristan: I know.
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!
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:
In one of the Evil Council episodes, they manage 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 a 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 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.