Arguably, any time Lightning and Serah talk about or to one another, and whenever old party members from XIII show up to help save the day.
Noel's speech about his motives and what he wants at the start of the game. He wants a future that has people in it, given how bleak and depressing his time is and how he was the last person born and the last person left after everyone died.
The ending starts this way, with Fang and Vanille safe, and the possibility of Serah finally being reunited with Snow and Lightning. Then, of course, everything goes to hell. But up until that point, everything was awesome!
Noel reveals to Caius that the reason Yeul keeps up the process of reincarnation is just to be with him again. To be clear, Caius isn't a Nice Guy, but damn if you don't kinda feel for him during his breakdown.
A sidequest allows you to find out more about Mog and where he came from. Apparently he'd been playing with his fellow Moogles until he got caught up in a time paradox and lost his memories in the Void Beyond until Lightning saved him. You eventually get to a time gate where he remembers everything that will allow him to return to his time and his own kind, but that means he'll be leaving Serah and Noel and their journey behind. He chooses to stay with them.
Mog: Mog has a new home - right by Serah's side, kupo!
Serah and Lightning's final conversation in Valhalla shortly after Serah's death. Not counting the mostly informational meeting in New Bodhum 700 AF, this is the first time they were able to talk personally in years, and Serah's speech to Lightning about how she's happy and doesn't care that she's dead is very emotional. Serah happily snuggling with Lightning before disappearing makes it all the more touching.
Sunleth Waterscape, 300 AF. After fighting a pack of Miniflan for the Thundering Artefact, Snow and Serah discuss her crystal stasis from XIII. Serah confirms that sleeping in crystal means quiet pleasure until you wake up, and Snow takes comfort in the fact that Vanille and Fang have been spending three hundred years in peaceful dreaming. The real kicker is the Live Trigger when Noel makes to speak up. Two-thirds of the game's Live Triggers have a 'comedic' option that acts as a massive Moment Killer, and this one has the very out-of-place "Feel like a song?" on the bottommost button. In gameplay, that counts towards the 'comedic' option for your adornment reward... but in the cutscene, Noel just suggests that Snow sing to Fang and Vanille to let them know he's here. Snow doesadmit that his singing would give them nightmares, but it's not a blatant attempt to force comedy into the scene; it's just him caring about his friends' dreams.
Before the revelation of the Proto fal'Cie, Academia 400 AF puts previous cities on Cocoon to shame. Noel is absolutely breathless at the sight of it, especially after Serah explains just what it takes.
Noel: Wow. How do you build something like this?
Serah: A little bit at a time, I guess. It probably took a lot of people working over decades, maybe even centuries, to make this city what it is now.
Noel: They had a vision for the future, and it grew from there, huh? You know what, Serah? I used to think that the only way to make a better future was to change something right away, in the here and now. But creating something for other people to build on, that's not a bad idea. Right? Even if it can't be completed in one lifetime, it's still worth starting. If you believe that someone in the next generation will take up the same ideal, then the dream lives on. [...] I wish she could have seen it.
A sidequest in the Vile Peaks has you locate six soldiers crashed there because of a paradox. When you have located Blitz, the sixth and final soldier, he immediately recognizes Serah as Lightning's sister. When asked how he knows Lightning, he explains that he and his buddies were once Lightning's colleagues in the Bodhum Security Regiment. Lightning's sudden resignment and subsequent rebellion against the Sanctum ticked them off at first, but they eventually came to understand that she tried to save Cocoon and, inspired by her brave actions, they decided to apply for positions on the Academy's force. They named their unit the Blitz Squadron in her honor and some of the members even took on a moniker related to Lightning. To seal the deal, Blitz says that while she may be gone, he believes in his heart that he and Lightning are still on the same team, trying to make the world a better place... It's only a short conversation in a big game, but it's a touching one.
A minor one: return to A Dying World after freeing Noel from his dispairing dream world, and walk all the way back to the empty settlement. The entire area is completely barren and empty, but when you get there you find a herd of wild chocobos - not the paradox kind either. Normal, yellow Pulsian chocobos. Immediately after that Noel renews his conviction to save the future. It's minor and optional, but it's definitely a relief for the player, who at that point has just played through the bleakest level in FF history.
Several Paradox Endings provide nice little moments to make up for the real ending's absolute heartache.
"The Future is Hope" prevents the most heart-wrenching moments in the game from ever happening. Snow get the person responsible taken care of, make sure Hope is safe, and gets a lasting reunion with Serah. It ends with the two of them riding Shiva into the skies of Academia on a brand new (yet related) adventure through time.
"Beneath a Timeless Sky" is more subdued. Sure, Serah and Noel are trapped in a desolate world, but Yeul is with them and she doesn't have visions anymore so she's safe. The overall mood is one of hope: "things are bleak, now but they'll get better."
A low-key element of Heartwarming: after realizing that there are many different Yeuls, Serah and Noel treat each Yeul as her own person. Yes, the fact that they all share the same name and body and they can all see the timeline means that they're usually a greeting by name, and yes, is the matter of "she is the seeress, so there are certain things that she absolutely will and will not do". But each Yeul is treated differently; Serah recognizes four different Yeuls as different Yeuls during her loneliness in the Void Beyond, and after the final boss, when they're discussing Caius' motivations, they refer to "them". Not "her", "them". They recognize that no matter how identical each one looks, they're different Yeuls.
The DLC Sazh's Story: Heads or Tails? starts off with Sazh flipping a coin, with the slow-mo title shot displaying the two faces: the goddess Etro (Heads) and a dark skull (Tails). Time freezes, he gets a bit of exposition from the casino owner, he catches the coin, and then flips it again because he didn't get a chance to catch it properly. Slam, peek... "What the...?" And it's a chocobo. Cue Chocobos of Cocoon in the background as Frocobo flies up to him. That falls under Funny.The Heartwarming kicks in when he amasses enough Fortune Medals to reunite with Dajh. That is beautiful enough, with them reenacting the Nautilus scene from XIII and Dajh NOT crystallizing, but then the casino owner tells them that "the possibilities have begun to disappear". Mystery Man calls a couple of coin showers onto the Chronobind table, and they're all coming up tails. The skull. "The possibility of the coins coming up heads has disappeared from the world. [...] The future has begun to disappear." Sazh then comes just shy of quoting Vanille: "Even if I my actions won't make a difference, I have to at least try." He grabs a coin and flips it, slamming it against his hand with Dajh and Frocobo watching... then he peeks, and a smile rises on his face as he reveals the coin to the owner and Dajh. It's another chocobo. If that doesn't scream Take a Third Option, I don't know what does.