Despite how Narmy it was, the scene of Dante crying over Trish's death is remembered by several as the most touching moment of the series.
When Trish tries to apologize to Dante for selling him out to Mundus, Dante comforts her with his motto: "Devils Never Cry."
Trish: Dante, I... I...
Dante: Trish, devils never cry. This tears, tears are only what humans have.
Dante just seemed so happy that Trish was all right. She practically glomped him in the ending.
Devil May Cry 2
Late in Lucia's disc in 2, Lucia is still distraught about her real origins, but Matier reassures her that the value of their history and experience is far greater than the circumstances of Lucia's birth.
Matier: Now, go. Everything that I know, I have passed on to you. You are my daughter.
Lucia: Thank you, Mother.
Dante wiping away Lucia's tear and telling her "Devils Never Cry." when she volunteers to go on what may be a suicide mission, wishing to die like a human. And when it seems like he goes into Hell instead of her based on a coin flip, the end reveals it was heads on both sides the whole time. Even in this game, Dante's still a really great guy.
Devil May Cry 3
"Quite frankly, at first, I didn't give a damn. But because of you, I know what's important now. I know what I need to do....Trust me! I'll make things right for you. That's what my soul is telling me to do."
Dante crying over Vergil's apparent suicide/death...followed by Lady realizing that Dante doesn't want to explicitly admit it, and handles it tactfully with "somewhere out there, even a devil may cry when he loses a loved one," thus naming the series. Dante even replies with as close to an admission as it gets, with a "maybe."
Devil May Cry 4
Out Of Darkness is this in itself. The song sounds like a love letter to Sparda, considering how he's the focal figure in the Order of the Sword's belief.
Unusually for the series, Devil May Cry 4 features a genuinely, if quietly, moving relationship between the hero Nero and his lover Kyrie. Halfway through the game Kyrie is kidnapped by the Order, who intend to trap her and Nero inside their artificial god and use them as a power source. Nero's eventual rescue of Kyrie, almost completely ignoring his hated enemy to catch her instead as she falls, is perfect.
Credo is gravely injured and at death's door and collapses as Dante catches him. Despite his fighting Nero recently and his friendship with him forever ruined, he begs Dante to save him and Kyrie (Who has also been captured) right before he dies. Dante agrees, saying he wouldn't want to dishonor someone's dying request, even though we know he'd do it anyway.
Dante giving Nero the Yamato in DMC4's finale:
Nero: What...? I thought this meant a lot to you...?
Dante: That's the only kind of gift worth giving. I want to entrust it to you, and so I am. What you do from here is your call.
Nero: Kyrie... If I'm a demon, and not a human anymore... is this what you want?
Kyrie: Nero, you're you. And it's you I want to be with... I don't know anyone who is as human as you are.
Seeing Dante working with his friends with familiarity really highlights the difference between him and most other top-tier demons/demon-hybrids: he doesn't work alone or horde all of the power to himself. Not only does he entrust power to others (like with Yamato), he actively encourages them to be paragons themselves.
Dante and Nero working together somewhat fulfills the What Could Have Been longing in the fanbase for more of Dante and Vergil working together. It helps that Nero's personality is basically a fusion of Dante's and Vergil's best traits.
A lot of the anime had this, from Dante dealing with/caring for Patty and the scenes between him and Lady and Trish. Probably the most heartwarming was the very end where Dante comes out alive after his battle with Abigail and everyone is just overjoyed.