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Tear Jerker: Fate/EXTRA
The Prologue has a nameless character's senseless death. Particularly how, as he lays dying, he wishes that somebody, somewhere, will remember his name...
Possibly worse when you consider that it almost sounds as if he's addressing the player... but you never do learn anything about him.
And it's not just him: The sheer number of bodies around him means that a lot of people died terrified and alone, with no memories and nothing to comfort them.
Pick any of your first three Elimination fights and someone will likely find it tearjerker material, be it finding out that Shinji is just a kid who didn't know any better, or the 'real' Alice accepting that playtime is over, especially if you said you cared about her, or Dan and Robin Hood finally understanding each other just before they fade.
Possibly even your character's reactions to the above. Unlike the other contestants, the player-character is appropriately and believably shaken up after essentially murdering their opponents and then watching them die.
The aftermath of Alice's Elimination Battle is especially painful, as you are, essentially depriving a little girl of her wonderful afterlife after a horrible, lonely life in a hospital in the name of your own survival. The protagonist and Nursery Rhyme were the only ones who played with her, the latter being summoned by her just to have a company. And Alice thought is was just a play and they were never hurting anybody, yet she knew that it will all disappear soon. And to twist the knife further, she asks you if you cared about her, and you can say no if you have no soul. And she comforts Nursery Rhyme as she breaks down in tears. It's hard to not feel like crap after this, especially as the protagonist is crushed by her true death far more than 2 earlier ones, telling himself/herself that (s)he can't desensitize himself/herself to killing and calling the system pointless and twisted to the core. Even later in the game, (s)he states (s)he will regret having her blood on his/her hands and not talking to her more when (s)he had the chance until the day (s)he dies.
Right after Alice vanishes, Nursery Rhyme notes that she doesn't have a true form, only that of whoever summons her. Despite that, she's happy that she was summoned as Alice, and laments that she will never be her again.
Some of the backstories revealed in completed Servant profiles. Robin Hood's in particular revealed that he'd done his duty to protect the villagers despite reaping only scorn for his unorthodox methods, finally dying and being known as anything but a hero. Despite his cynicism, he wished for a chance to be a true hero and be recognized as the same. It makes his final lines that much more effective.
Related: His post-match dialogue with Dan.
Archer: ... I'm sorry, Master. I knew I wasn't cut out for a fair fight. You needed someone better than a charlatan like me.
Archer: ... I'm so sorry. Another Servant wouldn't have failed you like this.
Dan: No, Archer. I'm the one to blame.
Dan: ... Thank you, Archer.
In one route, seeing Arcueid as a Berserker can be jarring to those familiar with the character. And just in case that doesn't drive home how Blessed with Suck Berserkers are, eventually Assassin gets to Come Back Wrong as one.
The former not so much if you know her backstory. Or played Melty Blood
Julius' final death, regardless of whether you wind up holding his hand as he fades or simply shedding tears he can't even brush away. Witnessing his Start of Darkness just prior ramps it up.
Not to mention the implication that the flashback is of the day that woman - his and Leo's mother - was killed on the Harway family's orders. It's further implied that Julius was sent not just to escort her but to kill her. And on top of that, her last words are to protect Leo, with no mention of the burden it put on Julius or even consideration for the fact that he'd undergone horrible surgery too.
That one NPC Master from your classroom who, unlike the others, seems to genuinely consider you a friend; the two of you can even promise to "go all-out" if you get matched up, and he says he'd expect nothing less of you. Of course, one day he disappears too.
There's another NPC in the classroom that is genuinely distressed at the idea of killing other Masters.
The end of the seventh Elimination Battle. In his defeat, Leo realizes what a fool he's been. He had never experienced defeat, which meant he lacked understanding of a vital part of living— So he could never have become the king he thought he already was. He then apologizes to Gawain for making him serve an incomplete king...and thanks him for helping him learn the final lesson he needed to know. What further drives it home is the protagonist's thoughts — he/she is reminded of mayflies in how they get their wings upon reaching maturity, then only live for a day. Poignant because now that Leo has learned his final lesson, he can't improve himself with it — because he's about to die.
All three of the playable Servants' back stories are pretty sad.
Part heartwarming, part nightmare fuel, and part tearjerker: after Archer (the playable one) realizes in week seven that he places the player character's safety over the Moon Cell, he makes an almost throwaway comment that if the PC had the Command Seal to spare, he'd force them to use it to change his class to Berserker. (This after having previously speculated, in another offhand comment, that Berserkers might be exempt from the Moon Cell's requirement that Servants prioritize its security above all else, and their Masters might thus not be subject to punishment by SE.RA.PH for their Servants' failure to adhere to that requirement.) What the hell, Archer.
One of Archer's last lines, particularly if you've gone through the Unlimited Blade Works route of Fate/stay night:
Archer:"Leaving the future to your real self once you wake up, huh? I see... I guess we're more alike than I thought."
Or, for that matter, Archer's speech to Twice Pieceman. A deluded, spiteful ghost hell-bent on destroying the present in the name of his misguided ideals, no matter who he had to sacrifice? Can you give a Take That to your alternate self?
In CCC Route Ending, when you permanently lose your Servant, especially when your Servant bid their farewell to you before fading away (except for Gilgamesh, who is implied to still be alive) . But, what's even more heart wrenching to watch is when the Protagonist ran to the barrier and kept banging on it, desperately trying to reach out for their Servant, but to no avail.
This is much worse if your servant is Caster during the CCC Route. She loves you and has her heart broken because you love Sakura and she can't escape with you. She had to put up a facade of happiness after panicking to tell her master to go. Saber also breaks down a bit, but is able to keep her composure much better.
Gilgamesh's shared dream of his life from childhood to the end of battle against the Bull of Heaven, told from Enkidu's perspective. Hearing the usually arrogant voice broke halfway through his speech was painful.