Let's face it: The original war was awesome, but how many got any happiness out of it? So seeing the Masters and the Servants peaceful, happy and loving each other is impossible to frown at.
Shirou and Rin's "cat-pajamas" sex-scene... or more accurately love-scene. For a refreshing change in the Nasuverse it was a moment where both parties took joy in the act, and the absolutely adorable smile that Rin wore through the whole sequence will warm your heart as much as set it racing.
That Caster and Souichirou are married after all the things they've been put through in the original game is heartwarming in itself.
We also get a look at Souichirou's feelings on the matter. While he doesn't really understand his own feelings on the matter (being basically emotionless for his whole life), he actually does have feelings thanks to Caster. And judging by his actions, they aren't bad ones.
Caster and Kuzuki's backstory is a mix of Heartwarming and devastating Tear Jerker, but possibly the warmest moment is the recount of the first time Kuzuki, raised as a Human Weapon ever since he was a Tyke Bomb, meets Caster... and his heart, that had beat the same regular beat for ten years without variation, skips a beat.
In the Hanafuda minigame Caster is shown making a lovely little scarf for Souichirou. And in their winning sequence he puts it on.