In Volume 1/Episode 9, when Klein asks Asuna to take care of Kirito. It's played for laughs in the LN, but is heartwarming in the anime.
Chapter 19 when we finally get to see the story behind Asuna and her actions with Kirito and damn is it full of fluff from their accidental meeting and to their in-game marriage.
Episode 4 with Kirito and Silica with Kirito talking about his sister and how he's been so aloof towards her once he found out they're not biological siblings and feels so bad for ignoring her. Silica tells him that she's sure that his sister still loves him anyway which ends up making him smile.
The last parts of Chapter 15 and up until 16 and its aww.
In Volume 2/Episode 3 "Red-nosed Reindeer", Kirito intends to solo a level boss to get a resurrection item to revive the dead Sachi, but is slipping deeper into a Death Seeker phase. Klein shows up, with the rest of his guild in tow, and begs Kirito to party with them and challenge the boss together, because Klein is afraid he'll die if he tries to solo the boss. When a PK-happy guild shows up, 30 men strong, Klein leads Furinkazan's 10 man guild to Hold the Line and give Kirito a chance to beat the boss.
After the boss is defeated, and Kirito learns the resurrection item won't work for Sachi, he gives it to Klein.
After that, during the Christmas of the first year in Sword Art Online, Sachi's recorded message for Kirito - obviously pulls double-duty as a Tearjerker.
Agil's wife kept his cafe running, and waited patiently for him to come back to her. He did.
The entire chapter/episode focusing on Kirito helping Silica revive her pet dragon Pina is just filled to the brim with this.
In Volume 7, overlapping with Awesome: Kirito and Klein arrive to Hold the Line against a rival guild, so that the Sleeping Knights and Asuna can challenge the level boss.
In Volume 7, Ill Girl Yuuki logs into ALO one last time as she's dying, to say goodbye to her friends. And then the leaders of the main factions of ALO perform a fly-past with their members. Over ONE THOUSAND ALO players from different factions put aside their differences to give her a send off.
Beyond that about two hundred players come to her real world funeral, surprising her remaining family who thought she had no friends.
Volume 4's denouement. Kirito and Leafa dancing together in the sky, and coming to terms with their emotional baggage.
For the anime, the end of Episode 10 where Kirito asks Asuna to marry him is chock full of this, and has a very powerful When She Smiles moment.
Episode 14, where Kirito and Asuna finally tell each other their real names. And when Kirito wakes up in the hospital, weak from two years on life support, he immediately gets up to find Asuna.
Episode 24: Kirito and Yui finally reach Asuna's cage; hugs and tears all around. After Sugou's dealt with, Asuna and Kirito embrace once again; she calls him her hero and while he says he's not one, he'll try for her sake. Asuna logs out and hopes that Kirito will be the first thing she sees when she wakes up. When Yui returns, (having been banished by Sugou) she hugs her papa and then asks if her mama is okay. Even Kayaba contributes by giving Kirito a Heroic Second Wind and a I Was Just Passing Through excuse.
The last twenty or so minutes of Episode 25, so much. Kirito and Asuna finally meet in real life. Asuna's hearing is off because of her prolonged dive but she understands what Kirito is trying to say. The next scene sees them in school, eating a lunch Asuna made, then sharing in minor PDA. The screen pans out to Lizbeth and Silica with their I Want My Beloved to Be Happy. THEN there's a party for the survivors of SAO that celebrates Kirito clearing the game. Back inside Alfheim, Leafa shows Kirito how to dance inside the game and then everyone (Asuna included, who is now an Undine) flies toward the reborn Aincrad.
Also making an appearance are Thinker and Yulier, who have married.
In Volume 9, we get this from Eugeo: "Whe-When we were young…we promised…me, Kirito…and Alice, from the day we were born together, to the time we die together…this time, I must definitely…protect…I must…"
A small but meaningful detail: In Episode 1 of the anime, there's a scene of a G.I.R.L. and a young looking man revealed to be a skinny man and a stout somewhat older man when Kayaba resets all the player avatars to reflect their true real-world selves. A few episodes later, they're seen talking together. At the end of the Aincrad arc, there's a brief scene in a hospital where these two men reunite in the real world with much happiness, to the surprise of the hospital staff. Whether or not this included a Relationship Upgrade at any point is never addressed.
Episode 23: after finding out who Kirito is, Leafa challenges him to a sword fight. In the middle of the fight, she throws a fight, but he did the same as well. Then, they hugged each other and apologize for keeping their true identities a secret from one another.
Once Kirito enters into ALO he checks his inventory to make sure a very special item is still there. When he selects it Yui appears and he embraces his daughter tightly in their first reunion since SAO.
Once the ALfeim arc is finished, the extra addition follows the SAO crew going on an underwater quest so Yui can see whales. It's very touching the lengths they're willing to go in order to give the little girl a fun experience.
Episode 7, Season 2: Kirito is telling his nurse Aki about how he killed three people in SAO and only now feels the guilt for it due to having encountered one of the survivors of the group they were part of. What follows is Aki reassuring Kirito that he did the right thing and that he deserves to move on from the guilt.
Episode 13, Season 2: Sinon giving Kirito a big hug as she blows them both up with a grenade. Coming after everything that's happened, it really feels like everything will be alright ...until Kyouji tries to kill Shino.
The ending of the Phantom Bullet arc. Kirito meets with Sinon in the real world and brings her to Agil's bar (after a meeting with Kikuoka). There she meets Asuna and Lisbeth who Kirito told about Sinon's past, so they could track down the bank teller who the robber was threatening that day. It turns out, she was pregnant at the time and she and her then-unborn daughter wouldn't have lived if it weren't for Sinon doing what she did that day. The clincher is the picture that Mizue (the daughter) gives to her.
Shino onee-san, thank you for saving mama and Mizue
While this is already heartwarming enough, episode 14 of SAO II makes it even more heartwarming.
Caliber Epilogue: Kirito builds a system to let Yui interface with Agil's bar.
Episode 18, Season 2: The girls are all at that little house on Floor 22, at a sorta-sleepover, and a few of them are feeling sleepy. Why? Because of the adorable sight of Yui, cozily sleeping on the resting Pina, who is also resting on the asleep Kirito.
Then, in the flashback, the family's absolute tearful joy at finally getting their house back. Asuna said that it felt more like home than her one in the real world.
When they arrive at the purchase panel for the home, Navi-Pixie Yui flies out of Kirito's pocket and off-screen behind them; there's a flash of light, and then she drops down in her child mode and hugs them both from behind. The sight is enough to make a grown man squee.
Episode 23 is literally nothing but these. First, by using a setup similar to the one that allowed Yui to interface with Agil's bar, Asuna essentially brings Yuuki to school with her. What really takes the cake is how all of her teachers and classmates don't even bat an eye at this, and welcome her with open arms. Then, Asuna takes Yuuki to the home she used to live in, as she wanted to see it one last time before she died. The day ends on a bittersweet note; Yuuki thanks Asuna from the bottom of her heart for the wonderful experience, but it becomes increasingly clear that she doesn't have much time left. However, that's not all that happens. Encouraged by Yuuki's words, Asuna finally wins her mother over, by taking her inside the game and showing her some scenery that brought back some memories of her parents' home. After the fact, while she still seems to have her cold exterior, she decides to allow Asuna to continue going to the school she's currently attending, as opposed to forcing her to transfer as she initially intended.
Her mother, who was so icy and strict about everything and tried to keep a professional and distant attitude even with her own daughter at all times, breaks down into tears from the pleasant memories of her parents thanks to Asuna, who doesn't even hesitate to comfort her despite resenting how she's treated her.
Episode 24 comes the most Tear Jerker and Heart Warming moment of the entire Season 2 where thousands of SAO players pays their respect bidding Yuuki farewell after Yuuki logs into the Medicuboid one last time. It was very heartwarming that viewers would break into tears after her long battle against AIDS.
The episode also skips the opening for a montage of Asuna and Yuuki spending their days together making it all the more heartwrenching when Asuna learns that she's dying.
Before Asuna duels with Yuuki, Liz notes that Kirito wasn't Dual Wielding when he fought, and takes that as a key indicator that he doesn't fight seriously unless it's no longer just a game. An episode or two later, the Sleeping Knights go for their second attempt at the floor boss and they get stopped by the raid squad - a couple dozen are guarding the doors, and a couple hundred coming up from behind. Kirito comes out of nowhere to stop the crowd, and when Asuna gets the Sleeping Knights to deal with the vanguard, Kirito puts on a second sword. And not just "a second sword", but frickin' Excalibur. This is beforeThe Reveal that the Sleeping Knights are all terminally ill Medicuboid users, so for all Kirito knows, it is still just a game, which means that he's getting serious for the express purpose of letting his girlfriend and her new friends do what they're trying to do. Love at its finest.
A slight one outside of the SAO universe: in the fighting game Dengeki Bunko Fighting Climax, both Kirito and Asuna are playable characters. The game shows that even in another universe, they still love each other and the two are even shocked at the revelation of fighting the other if you decide to pit the two of them against each other. However, the biggest heartwarming moment comes when Asuna uses her strongest attack... which is Yuuki's Mother's Rosario. Even after she's long gone, Asuna still continues to remember her friend by using the move she entrusted to her. Now THAT'S true friendship.
Asuna explaining to the fisherman they befriended just how much meeting Kirito had helped her.
Asuna: When I began to think about Kirito-kun whenever I would fall asleep, the nightmares stopped.
Beating the Hollow area story means Philia can rejoin the players on floor 76+.
Just the simple fact that Kirito and the rest of the gang can just hang out and play a game together without the fear of dying, seeing as in this timeline they were in SAO for even longer than usual.
Seeing Strea happily play with the rest of the party is this considering what happens to her in the climax of the previous game. She's alive and is no longer alone, surrounded by people who care about her and she cares about. Even in the amusing side stories where she makes outfits for the female members of the party, the only thing she cares about is seeing her friends' smiles.
The retelling of the Mother's Rosario arc is just as heartwarming as it is in the light novel and anime, except this time there's one more silver lining: Yuuki's condition improves, meaning that she'll be able to stay with her new friends in the long run. Yes, Yuuki survives in this route.
The scene where Rain and Seven, two estranged sisters, reunite for the first time in years. Then they later perform a duet together at one of Seven's concerts.
The sisters' bond in general, particularly when Aiko selflessly insists that Yuuki be able to test the Medicuboid, and with it, get a clean room and a better standard of living.
Yuuki's Character Development, particularly her epiphany around the time she, her sister and Merida plan on forming the Sleeping Knights.
She did not know how much time she had left. But if her life was to be shorter than others, she would only need to run on accordingly, at a faster rate, with a larger stride.
Merida snaps her SAO memory card in half as a birthday gift to the sisters, signifying that she's abandoned her desire to try to enter SAO and risk her life.