The entire series, really. Considering how absolutely horrible the Nasuverse is and how dark their lives were seeing the cast just having happy days together warms the heart. Well, everyone except Lancer, but even he gets his in the end.
Also the presence of the Neko Arcs in general, they are so cute at the same time funny in their scenes that helps the series being a Breather Episode of the Nasuverse and that even child audiences could enjoy them (despite the overall franchise being based onEroge) or at least mature audiences who find their inner child to enjoy them.
Caster gets the first in the series by her complete schoolgirl Squee! at Kuzuki's reaction to seeing her in her tennis outfit in episode 1.
In Episode 3, while sitting in his class, Shiki sadly muses on being lonely and disheartened due to not seeing much of Arcueid recently. Despite him being portrayed in CP as a somewhat selfish playboy, it's obvious he still cares about her.
Then there's Arcueid herself willfully suspending her powers and enrolling into the school via normal means just to be with Shiki.
In Episode 8, Saber goes through a hellish workday at Ahnenerbe, including, but not limited to: Berserker assaulting the store, Gilgamesh being Gilgamesh, and people wasting food. At first, she gives the impression that it's because she wanted to help out (and to pay rent), but at the end, it turns out she wanted to buy Shirou a birthday present.
That also puts her motive into perspective. When Neko Arc Chaos asks her to return and work full-time, she snaps and asks, "Is that something you would ask of a king!" and he backs off. However, she willingly chose to work that one day for Shirou's sake.
Even better? Saber's Superpowered Evil Side still manages to buy him a gift while drunk on a trip in the town, and immediately transforms back into normal Saber after seeing him.
The entire Caster and Souichirou segment, culminating in Souichirou proposing to Caster as she uncontrollably weeps tears of joy.
It's even more heartwarming when, in almost every scene Caster is in has her acting like a dutiful housewife. With the above scene, it culminates even further with the racing episode they're in the newlywed car and they outright throw the race for the Holy Grail and leave because she says that she's already got what she wanted and then in episode 12's final moments where she's petting Len on her lap sitting next to Souichirou, but you'll notice that they're wearing Hawaiian outfits which means they either got their honeymoon or are about to go on it. Either way, you can't help but be happy for them.
And if you take Fate/hollow ataraxia into account, the segment in episode 7 might very well be canon.
There is also the fact that he catches her in the "disgusting" act of building Saber figures and he doesn't care. In fact, he praises her work!
Doubles as Brick Joke status in episode 9 when Caster and Kuzuki ditch the race to go on their honeymoon and Caster declares, "I already have everything I need!"
It's sweet of Arc to go to such lengths to find the finest ingredients for Shiki's okonomiyaki, even though Hilarity Ensues as a result.
Shiki's reaction to seeing her work that hard for him (plus how he comments that she looks cute in an apron) caps it off.
After Saber delivers the finishing blow to BerSer-CAR in the Ilya Castle Special episode arcade game.
Shirou: "Saber, we're having steak for dinner tonight."
Saber: "I love you, Shirou!"
This is after Shirou had punished Saber by telling her she'd get no meals for two days after falling short in the previous levels.
The post-credits scene in the last episode, where everyone's gathered in and out of the Ahnenerbe for the closing party, including Kiritsugu, Irisviel and Ilya having a family dinner (Ilya's contented sigh really sells it), Caster petting Len, and Taiga, Gilgamesh and Iskander chugging down beers. It almost makes up for the happy endings they could never have in canon.
And, among the ones celebrating outside, a smiling V/V and Avenger. Bet you didn't expect that one!
Special props to several of the heroines in episode 12. Despite Shiki and Shirou running around all over town as part of their Zany Scheme, the girls do show some heart-warming moments of contentment:
Hisui is happy to see Shiki after she waited for him for the past four and a half hours! Her voice completely sells it.
Arcueid's excitement about getting to spend the day watching movies; one of her favorite activities.
Sakura telling Shirou how much she'd been looking forward to their date.
Rin getting scared during the movie and reaching for Shirou's hand.
Even Sion's little moment when she catches up with Shiki. She's not angry and had literally been looking for him all day just to go on a date.
Finally, Kohaku's excitement at seeing Shiki return home safely after having waited for him all day in her lab.
The scene from the Tsukihime segment of episode 10, where Shiki is carrying Akiha on his back. This being Carnival Phantasm, few seconds later his mouth gets him in a lot of trouble with her.
From the EX Episode, Has to go to Lancer and Bazett... though Caren ruins the moment. You really wish Caren would do the right thing and just give Bazett back her command seals.
Hibiki provides a couple in the HibiChika special as a result of her cheerfulness: putting Chikagi into full dere mode just by asking what she wants to eat and considerably brightening Satsuki's mood when she helps her realize that the AAA still has fans.
And also concerning the Hibi Chika special, how does the entire thing end? With the usual credits scene, except Rider, Sabers Blue and Red, Kohaku and Akiha are all running together... And then the credits scene ends with nearly every single character from the Tsukihime and Fate franchises running across the scene, showing how the little high school scribbles of Kinoko Nasu has become a masterpiece franchise.
Saber's reaction to the vehicle she and Shirou get (a toy lion that needs 100 yen to work for a few seconds) for the Holy Grail Grand Prix in episode 9.
Likewise, Rider falling in love with the granny bike she and Shinji were given for the race. She even named it Pegasus Unit.
Despite the fact that Berserker drags him around all day and makes his life a living hell, Lancer doesn't seem to bear any ill will toward him in "Berserker's First Errand." They're seen sharing a beer together, Lancer calmly suggests Berserker just call it quits and go home, and even after Berserker smacks him down, Lancer calms down almost immediately, chuckles, and helps Berserker figure out how to complete his errand. Before Berserker grabs him up again, he seems downright happy that Berserker has everything figured out. Talk about good-natured. And while it is a moment of dark humor, it is actually rather touching that even after Lancer "dies", Berserker still goes back to get him and takes him home.
Kiritsugu, Irisviel and Illyasviel eating together in the cafe at the end of episode 12. Considering how things end up for them at the end of Fate/Zero, it's a really sweet scene.