Many Heartwarming Moments. Especially the True Ending.
Pretty much any post-Shadow scene has some of this, with the characters both accepting themselves and being accepted by the Investigation Team.
The final scene for any of the Social Links easily qualifies. Heck, there's generally several throughout any given S. Link.
It's not the final scene of his S. Link, but Yosuke's monologue in his penultimate Link is a fitting conclusion to his character development in its own right.
Yosuke:I want to tell Saki-Senpai something. That what's important... Isn't where you are. This town I hated so much? Now, I love it. There's still nothing here, but I have family and friends... and you. The important things are never far off... They're all around you. I always wanted to be 'special'. I thought my life'd finally have meaning if I was special to someone. That's why I was really excited when I got my Persona. But I really didn't need it... It's not what you have or what you can do... Just being born, living your life... Before you know it, you're already special to someone.''
"Yeah... Like you... You're special to me, you know?"
The penultimate rank of the Drama Club (Sun) Social Link also deserves a mention, when things turn emotional and not the regular "dramatic scene" music Reverie starts playing, but the epic I'll Face Myself, a song that is usually reserved for party members who accept their shadows. It makes an already great scene even better.
The follow-up to the end of Izanami's fight, the final boss in the true ending, is also quite heartwarming.
The Christmas Eve scene with a girl, any girl. Naoto's is a personal favorite.
Christmas Eve with anyone, really. Special note that if you don't have a girlfriend (or decide not to spend any times with any of the girls), Yosuke, Kanji, and Teddie will decide to have a bros party with you. Christmas eve with the dudes especially because it also doubles as one of the game's MANY Crowning Moment of Funny, and coitus does not ensue.
When Nanako comes back to life, if you choose to spare Namatame. Also when MC checks his phone afterwards, and his inbox is flooded with messages from all of his friends, offering support. Subverted if the player hasn't maxed out any non-party school social links by the time this event comes around. In fact, getting the narrator's "There are no new messages" is actually kind of heartbreaking.
The fact that Dojima still considers Adachi as his partner despite the things he did. Even moreso if you consider this may be at least part of the reason Adachi not only fully cooperates with getting himself convicted but even helps you find the true ending.
All of Kanji's social link is adorable. But when the little kid vouches for his innocence with the police officers, you just want to give your playstation a hug.
When the player first formally meets Kanji's mom, she reveals just how much Kanji appreciates his friendship with the protagonist.
Throughout the game, if you score highest in the exams, Nanako gives you some very useful homemade accessories. After the final exam of the game while Nanako's in the hospital, if you check the cushion she usually sits on you'll find a writing assignment by Nanako talking about how her big bro is the greatest guy in the world, as well as the final accessory.
And one could say the Heroic Sacrifice of everyone during said Izanami fight was incredibly heartwarming (and slightly heartbreaking).
And the trumpet that plays when Yosuke knocks the player out of the way of the Thousand Curses this video also deserves some mention, as it plays in almost heroic tune as Yosuke is struck by Thousand Curses.
Chie's Social Link, my favourite being the romantic route, always makes me fangirl inside.
Teddie's resolution to come back after realizing he's a shadow. And Yukiko finally laughs for the first time to a Teddie joke.
Ms. Kashiwagi's surprising friendship with Hanako, especially after the beauty contest. Yukiko later says that: "They often come here [to the Amagi Inn]."
Max out the Hermit S. Link and visit the shrine at the end of the game. You see that the Fox has a trio of adorable children.
When the protagonist is inflicted with fear status in battle, occasionally Rise will say, in a very sincere voice, "No, don't be scared, I'm with you!" Especially heartwarming when she says it and you manage to get over the status the next round.
It tends to be overlooked since she was only mentioned in passing but Elizabeth, left the Velvet Room to find a way to free the Persona 3 Protagonist from his Barrier Maiden fate.
If you refuse to date Ai in rank 6 of her social link you get a heartwarming moment in rank 9 in which Ai jumps in front of you to receive a slap that was meant for you (remember, Ai's life goal was being pretty). She then says that you've done so much for her that she felt like doing something in return.
After defeating Mitsuo, the Investigation Team has a party where the girls (and you) make omelets for Nanako. The girls' turn out... predictably, but Nanako says that all of them (even Chie's, whose came out the worst) are good.
Added in Persona 4 Golden:
Any girl that you are in a relationship with will now call the Protagonist up to ask if he'd like to go to the summer festival with her, to make up for Teddie's antics the previous day. Doing so, and then commenting "You're right, it was totally different" following the fireworks display at the end of the festival & Yosuke lamenting that Teddie kinda ruined the festival for them, leads to the girl blushing & nervously fidgeting as the rest of the group slowly puts two & two together.
Yosuke: "Dude, Seriously?"
Your date will invite you on a date to the shrine on New Years day, And if you draw from the shrine with impure thoughts you'll get Great Blessing for both you and your Girlfriend... Only if you are dating just her. or else you'll get "CURSED!!!"
The Ski Trip, You get to spend time with whichever girl in your party member and you two will end up alone in the cabin. However, you can choose "The guys".
Valentine's Day, you get to choose one lucky lady to spend time with at the beach. Assuming you don't decide to reject them and get called out on it by all of your girlfriends. Still, the girl you do choose is very happy, even if she's... done something unique to the chocolate she makes for you. Special mention goes to Marie if you max her social link: she'll be there at the Dojima residence towards the end of the day to give you chocolate whether you romance her or not even if you've romanced another girl. And if you did romance Marie, she'll take it further by ending the scene with:
Marie:I love you. I love you so much. *kiss*
After the protagonist decides to get his motorcycle license with Yosuke, Dojima decides to give his old scooter to the Protagonist to use whenever he wants. Best uncle ever.
The shrine visit on New Years with whichever girl you're dating. Unless you're dating more than one girl, then not so much as drawing a forrtune with impure thoughts results in the Protagonist drawing a curse, advising him to stick to one girl, and the date obliviously commenting that the fortune must be wrong. Special note to Chie, Yukiko & Rise's events, as the three of them arrive glammed up & in kimonos, something outside of Chie & Rise's comfort zones. In fact, it's so different to how Chie normally is, that she comments that her family were teasing her about it, so she just decided to tell them about her boyfriend.
Originally, when the game progressed from Christmas Day to the Protagonist's final day in Inaba, it merely went down the calendar rapidly. In Golden, the Protagonist meets Yosuke, Chie & Yukiko outside school the day after Valentine's, and upon reflecting that it's nearly spring, they promise to use the little time they have left creating great memories, before heading into school... As the music from the title screen begins playing whilst cutting to the calendar; except this time, the calendar quickly fades to black & the player is instead treated to a slideshow of sepia toned pictures of that final month, ending on the Investigation Team taking the photograph the Protagonist looks at in the original version of the True Ending.
With The new Adachi S. Link He'll cheer for you during the fight with Izanami in his own special way
" Man you're so cruel."
" You make me take responsibility for my sins, but you're gonna let this bitch take you down like this?"
" Haha... That's hilarious."
" ...Get up. You're not like me, right?"
The epilogue which features the protagonist returning to Inaba in the summer following the True Ending. Namatame's become The Atoner and, inspired by the Investigation Team, is running for political office so he can help change Inaba for the better; the business owners have moved past their resentment towards Junes, and both the local businesses and Junes are working together to revitalize the town; the rest of the Investigation Team are hiding from the protagonist as they're in the midst of throwing him a surprise party to make up for him not being able to relax when he visited in May; all seven of the protagonist's teammates have matured in appearance, showing that they've long since accepted their Shadows; Dojima found a lead in his wife's hit-and-run case, and credits it to the protagonist's influence on him; Nanako has taken after her mother and begun playing the piano; and the Dojimas and the Investigation Team throw a "Welcome Home" party for the protagonist. It's genuinely heartwarming to see the Investigation Team and the Dojimas in a much better place than when they were first met, and the final scene of the game is of the protagonist happily smiling with his friends and family around him.
A line in Adachi's letter from prison.
Adachi:When I'm here, I think of Dojima-san, Nanako-chan, and you. A lot. Though my time with you didn't seem like that much fun before... it's strange. Despite it all, I'm grateful to you. Thank you.
Every single third-tier Persona scene. Basically, each party member caps off their development with a heartfelt thanks to you and a resolve to continue to grow further and support you and their other loved ones. Kanji's in particular is adorable, where he enlists the protagonist to help him create a gift to show his appreciation for his mother.