Nils saying goodbye to his sister Ninian and Eliwood before he crosses the Dragon's Gate alone in Fire Emblem.
Kent and Lyn's A support: I am here for you. Even were I not a knight, even were you not my lady. My heart would not change. So, I hope that you will forgive me, should I continue to stay by your side...
Priscilla and Erk's A support, Say My Name end included. Even more heartwarming because it triggers Priscilla's only happy double ending, where she * does* get to keep the man she loves.
Serra's support with Oswin, which also showcases her desire for his approval and how crushed she is when she thinks she's a bother on him and the troupe. Or when Hector says she can consider him and the rest of Ostia her family for as long as she wants.
Lucius may be crowning character of heartwarming. In his Renault support, when he talks about his years of suffering after his parents' deaths, Renault confesses that he killed Lucius's father, falls to his knees in tears and begs Lucius to forgive him. Lucius forgives him.
In the last chapter when Eliwood tells everyone to stay back and, of course, no one listens to him.
Also, before the chapter when Eliwood talks to Nergal: "Do you believe for a moment that, when wise Athos struck down his dearest friend, he felt nothing!?"
Lowen and Rebecca's supports as a whole are pure heartwarming.
Jaffar/Nino's support: Nino showing her trust for Jaffar even after his job was to kill her and Jaffar admitting that he loves her.
Lyn's reunion with her grandfather at the end of her tale. Hausen's reaction, Lyn's assertion that despite taking so much poison he'll recover, and the CG of them embracing is, in this troper's opinion, one of the most heartwarming moments in the game. Also a bit of a Tear Jerker.
In that scene, when Lyn talks of the pain of being alone after the Lorca massacre (the less-than-a-dozen other survivors all went their separate ways, refusing to follow her), Hausen solemnly apologises for his pride all those years ago: he reveals how deeply he regrets not accepting his daughter Madelyn's choice to marry for love, and wishes he could take it back because they could have all lived together happily. In response, Lyn assures him that despite the tragic end, her family and tribe all lived a life of love and happiness; Hausen, deeply moved and happy that Madelyn had a happy life, thanks her. This, in turn, leads to the above bit where Lyn assures him that despite how much poison he's consumed, strength of will can save his life.
Dorcas and Farina's supports in FE 7. Dorcas's wife has a bum leg and Dorcas works to earn money for her treatments, and when Farina finds out she gives him this huge sum of money. These supports are among the best for revealing Farina's Hidden Depths.
Hawkeye's supports with Ninian provide some information about Arcadia, a hidden paradise where dragons and humans live in peace together. At the end of their A support, he tells her he has hope for a future where all of Elibe can be just like Arcadia, and Ninian says she believes this as well.
Hector and Farina's A support. She's distrustful of nobles due to Ilian mercenaries being treated like pawns, and spent all of their previous support reflecting this. Hector spends the entire final support convincing an overworked and exhausted Farina that no, he doesn't see her as just a pawn and he's helping her because he wants to. Keep in mind that Hector isn't known for his finesse and gentleness when it comes to women. This also becomes a Crowning Moment of Funny when you see their ending CG◊.
Hector's decision to wield Armads even knowing the consequences. He doesn't care if he dies young, or a painful and bloody death at that. He just wants to help Eliwood. Actually, Hector getting involved in the first place all because of Eliwood.
In Genealogy of the Holy Wars's prologue chapter, the normally cold and harsh Lord Arvis admits to Sigurd that despite only being his half-brother, Azel means the world to him. Say it with me: "Awwwwwwwwwwwwwwwwwww!"
The manga adaptation amps this Up to Eleven. Azel and Arvis meet up after the Verdane arc and have a talk, in which Arvis tells Azel he loves him and asks him to always remember that fact. Then we get a flashback to young Arvis bonding with baby Azel after their father's death.
A couple romantic examples from Genealogy of the Holy War: Adean's chapter 4 conversations with Azel or Midayle or Tailto's convo with Azel, and the chapter 5 convos between Dew/Raquesis, Lex/Tailto, and Ferry/Arden.
Seliph rerecruiting Julia in FE 4. Not only does it inspire a feeling of "Awwww....", it's also part Awesome Moment with you knowing you've truly outdone Manfroy this time.
Leif killing Travant and avenging his father Quan in Seisen no Keifu:
"Father, did you see that?"
Rescuing the civilians in Genealogy of the Holy War. Not only do they give you a free level up, but you also get to hear their heartwarmingly sincere thanks for saving their lives. Especially from the children.
Seliph's conversation with the ghost of his parents after killing Arvis, especially with the remixed version of Sigurd's theme.
For Fire Emblem: The Sacred Stones, we have Eirika and Seth's A support, where Eirika confesses that when they left their Doomed Hometown, she briefly thought they could've run away, yet repressed to say it because she knew they couldn't... and before leaving, Seth admits that he thought the same as he held Eirika during their escape.
A non-romantic example from The Sacred Stones: Ross and Garcia's A support. Even after Ross called him out of retirement, Garcia remains a father first and a fighter second. Also their C support where they talk about Ross's deceased mother.
Yet another The Sacred Stones example is Forde and Vanessa's A support, specifically this quote: When you first saw me sleeping, I was dreaming of running across a field. I felt this warm energy engulfing me, blowing across my face like a summer wind. It must have been your energy that made me feel that way.
Another The Sacred Stones example: Franz and Amelia's A Support
Amelia: Let me be your shield to protect you...
Franz: And I will be your sword and fight for you. From now on. You're not alone anymore, Amelia.
Amelia: Thank you Franz.
And then she runs into his arms and they hug. Fluff overdose, here.
Myrrh is characterized as a humble and very lonely girl. At first she seems to prefer the solitude, but support conversations make it clear that she actually longs to have companions around her who she can be herself with. This wish is eventually granted in her paired ending with Saleh, who invites her to come live in his home village of Caer Pelyn. Instead of living on her own in Darkling Woods, she gets to live in the company of friendly people who, in time, learn to not treat her as the Great Dragon, but as a normal girl, bringing her great happiness.
Another non-romantic example: Innes spends his and Tana's B and C supports refusing to see her as more than his Tagalong Kid sister, but in their A support it's revealed this is his way of expressing concern and love for her, and Tana's love and admiration for her brother is made clear as well.
Artur and Lute's supports are as a whole heartwarming, but their A support is particularly so: after Artur confesses his love for her, Lute goes into a bit of a panic as she tries to recall if her books said anything about love. Artur then tells her that love doesn't need books, and that she should just be herself and listen to her feelings. After taking a moment to do so, Lute finally responds: "I'm glad... you're here."
In one support, Ross gives Amelia A necklace that belonged to his late mother
Fire Emblem: The Binding Blade: Roy and Lilina's A support and Roy's supportive behavior in chapter 8 after she finds out Hector's dead.
Clarine and Rutger's A support, wherein she shows him her soft side and he shows some of his own by admitting he likes her better in Well, Excuse Me, Princess! mode.
Roy telling a tearful Fae that she'll always be a friend to them even though she's a dragon, followed by his desire to help Dark Dragon Idoun regain her soul.
Noah/Fir A support is heartwarming as it is, but Juno and Noah's A support makes it even more heartwarming: Noah admits he hates getting close to people because there's a chance it will end badly, but he's gone out of his way to reach out to Fir anyway.
Gonzales' epilogue in Sword of Seals. Being recognized as a hero from a place where he was originally viewed as a brigand who would sack towns. All because Lilina saw that he was not a bad person at heart.