The ending has Tsukasa finally meet up with her beloved Subaru, the crippled girl who helped her through her imprisonment in the game. It's especially heartwarming since Tsukasa thought she was a guy herself for most of the series — but it doesn't matter to either of them.
Anything involving Tsukasa and Subaru is heartwearming, period.
Indeed. Another moment that stands out is when Subaru has just disbanded the Crimson Knights. She's kinda down and wandering through an area where she first formed the guild. A girl with a similar character design comes up and starts talking to her, then tries to Player Kill her. A beaten and tired Subaru wanders down into the cave where Tsukasa is sitting. Tsukasa initially panics, but she stops him, and they embrace. It's a very sweet scene.
Anytime the song "Grandpa's Violin" plays in the background, it's going to be a heartwarming moment.
The episode focusing on B.T.'s personal life. She seems to be bitter and lonely in real life, all the people in the game either dislike her for her cold and snarky nature, or merely want to use her as a way to get information. She realizes she doesn't have any friends. Then Tsukasa shows up and saves her from two Player Killers she was trying to be nice to, and she starts bonding with him.
Unison. All the main protagonists, including the ones from the games set after the events of .hack//SIGN, get together and have a strictly feel-good party, dancing the night away to the series' theme song.
Any exchange between Tsukasa and Subaru later on in the series.
Bear goes to extensive measures throughout the series to research Tsukasa in the real world. After BT semi-calls him out on using Tsukasa as a replacement for his own failed relationship with his real life son, he decides to take full responsibility and at the end of the series he adopts Tsukasa's player and puts her through college.