Red Saber; in musing about the irony of her trying to get rid of religion during her reign as Nero. And then comforting the hero/ine when s/he worries about his/her existence as a computer program. She, after all, is the magical construct of someone long dead as well; and she's sure that they will all be together one day "in the place beyond the Sun."
One of Saber's musings in the final Arena qualifies as well: she talks about the fact that while death may be beautiful, it is a beauty she cannot love, and wishes for your own death to be as peaceful as falling into slumber.
Say what you will about Leo and Gawain as enemies, but the scene after they're defeated, a flashback to Leo knighting Gawain as his Servant is touching.
Archer, as always, is a StoicDeadpan Snarker for most of the game. However, after he admits that he places your well-being above his duty to the Moon Cell, and you reply that you're willing to suffer the consequences along with him...
An actual, genuine smile lights up Archer's face as he repeats the word again and again.
"The bond that we share... Even if it's against the rules, I will not allow it to be destroyed."
Saber:"Praetor. You may call me a tyrant or even the Whore of Babylon, and I will not mind."
Saber:"Because - Because I like you!"
Saber:"It hurts, but if you dislike me, I will simply work hard enough to become worthy of your love!"
Saber:"O-Oh yes, I can endure this pain for you!"
Saber:"Just watch, one day I will take your heart!"
Vs. Dan and Archer, Saber provokes a clearly defensive stance from the usually-flippant Archer on Dan's behalf, causing Saber to state that she feels better about the match since Archer's regard for his Master is not so different than what she feels for hers.
Pretty much the entire Rani path, despite her initial coldness; even her mood after the lunch scene is a sign of her leaving Emotionless Girl status behind, and her belief that you'll return to have lunch with her again is so simple and steadfast that it's touching.
Rin's post-game dialogue as she vows to find and wake the MC even if it takes one of her infamous acts of violence. Then her voice goes soft as she adds
Rin:"And, if I can manage all that, I'd like our reunion to be on a rooftop somewhere... with a beautiful view."
Dan Blackmore's Archer didn't get much of a break in his past life As Robin Hood, and usually resorted to sneak attacks and poison. Due to Dan's chivalry, he issues a Command Seal to make sure he waited until the Elimination Round to battle. They lose, but manage to comfort one another in content; Archer was able to fight a fair battle, like a 'proper hero,' for the first time, and Dan would see his dead wife again, whom he wanted to bring back through the Grail if he had won.
When you have Gilgamesh as your Servant, he would make some threats of killing you at any time. But, as the story progress, there will be the time when the King of Heroes finally acknowledged you as his Master.
Gilgamesh: (to the Protagonist) I told you before that this fight is yours alone, but forget that. Your enemy is my enemy. I shall lend you a hand, nameless master. Henceforth, I shall be your sword!
In the true ending route you avoid the fight with BB entirely by confessing your love to her.
With Archer as your Servant in the CCC True Ending, even though it's tear-jerking as Archer can't come back to the Near Side with you, he reassured you that everythings going to be alright, because of you, he finally furfilled his childhood dream of being a Hero of Justice: Both you and Archer manage to save the world from the True Final Boss, Archer's given a chance to help those in need (You, the Protagonist), and finally save Sakura. He even says that he is now satisfied with everything they have archieve together to you in the very end and is quite touching and heartwarming considering Archer's dilemma.
Archer'sEnd(When the original protagonist finally awakens from the cryogenic suspension, and later meets Archer (before he became a Servant) in a refugee camp).
Gilgamesh End: As the protagonist was about to get deleted (following Fate/EXTRA's storyline) Gilgamesh appears and saves the protagonist from deletion. Now, two starts exploring to a new world (well, the Universe, to be exact) together, 1500 lightyears away!
As Shinji dies, thanks to his sacrifice in order to deplete Meltlilith's invincibility, he finally calls you, the Protagonist, his first real friend he ever had until the bitter end.
For Jinako's case, she began to get some respects and kindess from Karna, who likes her to the point that he changed his Luck stat from D to A because he feels so lucky to have her as a Master and is loyal to her that he sacrifice himself in order for her to live on; Monji, who gets along with her and even starts teaching her the meaning of God; and even from Shinji, that he was really impressed with her hacking skills that he starts to respect her.
If you ask Caster Andersen about your Caster, they quickly get in each other's skin and start trash-talking each other's physical traits... until your Caster notices just what Andersen's Innocent Monster had done to his image. Knowing what kind of story your Caster has, when she grows silent as she hears his monologue and refrains to blow him to kingdom's come, you know that they have a Not So Different moment. In Caster's own way.
Hakuno's interactions with some of the previously and currently enemy Servants are pretty heartwarming too: