Numerous occasions. Lancer's shock and grief at being betrayed, Berserker's final moments with Saber, Saber herself—there are too many to count.
Lancer's betrayal deserves a special mention here. In the previous episode, he'd just proven himself to be probably the most heroic Servant thus far, willingly giving up one of his Noble Phantasms and strengthening a future opponent to stop a kaiju from killing thousands of people who will never know his name. Then in this one, he was actually winning his duel with Saber—watch the positions when he and Saber have their weapons at each others' throats. Saber is already fully extended, but Lancer still has reach—that is, Saber can't do anything, but Lancer just needs to step back and extend his arms and he'd have stabbed Saber in the throat. Then Kayneth activates his command spell. What follows is the tragic screaming of a knight who just wanted an honorable battle and probably the best summation of the Grail War—nothing less than hell on earth.
Kariya Matou's farewell to Sakura after having just been infested by Zouken's insects in an effort to prevent Sakura from having to take part in the Fifth War.
"It seems I've already lost [a bit] to the bugs inside me. I guess I'm not as strong as you..."
Being The Stoic, distant, cruelly practical Magus that he is, you wouldn't think Kiritsugu would be able to inspire this with one sentence. However, his realization that "Ilya weighs less than that Walther [rifle]" and his reaction to this simply convey despair and depression and hopelessness far better than anything else ever could.
His breakdown in front of Irisviel in the second book is another one.
The prologue itself, too. Kiritsugu saying "Then, what about her? I... am not fit to hold that child", and crying at the realization that he's going to eventually have to choose between his beloved wife and daughter, and his ideals and his dream, fearing the time that he'll have to sacrifice what he wanted to protect.
Rider's incredibly blunt and harsh speech to Saber in Episode 11, as he systematically tries to break down her ideals, ending with calling her a little girl. Saber's shaken reaction and her flashbacks to the battlefield, where we see her broken and severely wounded, is definitely a tearjerker.
On the same topic, Saber after Kiritsugu forces her to destroy the Grail. She returns to the hill of her scene of death, even more emotionally broken and undergoing a severe Heroic BSOD at the thought of what the rest of her men experienced. She concludes that the fall of her country was all because of her, because she was chosen as the king, and starts crying and and crying and apologizing, over and over again, after deciding that the only way she can repent is to use the grail to erase all her mistakes.
Kayneth willingly gives up in the Holy Grail War in order to save Sola-Ui, throwing away his pride and choosing to protect the woman he loves instead. Then they both die.
Kiritsugu having to kill his mentor and adoptive mother to prevent a zombie outbreak in his backstory.
Depending on your timezone, the episode in question aired on Mother's Day for extra despair.
The moments before and after the final battle between Archer and Rider, particularly:
Rider asking Waver Velvet to be his "retainer," with Waver professing his undying loyalty to Broskander.
Waver:(Tearing up, his voice breaking even with a smile) You... You are... My king, sire. I will serve you, with devotion. Do me the honor of leading me. (Sobbing) Let me dream, what you dream!
The tense confrontation between Waver and Archer, where Waver stands up to Archer, proclaims his duty to Iskander to live, and earns Archer's respect before being left alone to break down and mourn over his fallen friend with Manly Tears. Doubles as a Crowning Moment Of Awesome.
The worst part? This episode aired on the day the real Alexander The Great died.
One of the last scenes in the fourth novel, Kiritsugu having his methods in saving people deconstructed by Angra Mainyu, which not only reveals how exactly the Grail is planning to grant his wish; it forces Kiritsugu to experience (and in some cases, re-experience) killing everyone he ever loved and cared about. The kicker, though, is when the Grail reveals what Kiritsugu's true, hidden desire is- to live a happy life with Irisviel and Ilya, a wish that he sacrificed to save the world... and then Kiritsugu proceeds to reject the Grail, which involves shooting the hallucination of Ilya in the head and strangling what's more or less an Avenger-possessed Irisviel, and "she" "dies" cursing him. His last words to her before she dies? "I will save the world". His last words to Ilya? "Daddy loves Ilya too... I swear, no matter what happens, that's true..."
And the episode which adapted that scene aired on Father's Day. Another Irony!
When Kiritsugu shoots the hallucination of Illya between the eyes and she falls over "dead," the Avenger-possessed Irisviel lets out a gut-wrenching, heart-breaking shriek of abject horror and sadness that just tears at you when you hear it. Illusion or not, the screams of a mother who has just watched her child die are nothing if not soul-crushing. Bridget Hoffman's delivery of the the scene is incredibly powerful, and leaves you feeling just... Just aweful.
Kiritsugu ordering Saber to destroy the Grail. Saber's scream is just heart-wrenching.
In the penultimate episode of the anime, Kariya's death. His body finally gives out from the strain of the worms and he just sort of... falls over. Alone in a dark basement, and no one really cares. The fact that he appears to be smiling makes it even worse.
The fact that Sakura - the person he fought so hard to save - called him a fool for all that he did makes it worse yet.
Kiritsugu going through the wreckage of the city desperately trying to find someone - anyone! - that's alive. When Kotomine sees him, he has a broken expression on his face.
Kotomine Kirei begging Kiritsugu to let Angra Mainyu be born. It comes off as Kotomine's last attempt to reconcile the path he's tried to follow with his own nature, before he embraces it wholeheartedly.
Not quite a Tearjerker, but Kirei's brief-but-unmistakable reaction when Kiritsugu looks right through him. For someone who rarely responds to anything besides physical pain and schadenfreude, it's telling that he doesn't derive any joy from seeing Emiya Kiritsugu broken.
If anything, it reminds us that Kirei, despite having fully embraced his inner nature, still retains his humanity.
No, he doesn't. He's angry at Kiritsugu for being broken; HE wanted to break or kill the man himself. Essentially, he's angry because, in his mind, Kiritsugu denied him his closure.
Aoi during the epilogue, believing that Tokiomi's still alive and Sakura's still a part of the family.
Tokiomi, after a heartfelt farewell in which he entrusted Kirei with, essentially, the Tohsaka's future, is murdered by Kirei, who's conspiring with Gilgamesh, and results in his wife suffering brain-damage. The anime precedes it with a Pet the Dog moment with Aoi and Rin, while the light novel has Tokiomi say:
Tokiomi: "To me, you are the best reward, Kotomine Kirei. With this, I can set forth for the final battle with no regrets."
The Azoth dagger, which had been a proof of love and trust, slipped between two ribs to pierce Tokiomiís heart.
The final conversations in the series, between Emiya Kiritsugu and Emiya Shirou, and the full conversation between Shirley and Kiritsugu from Kerry's childhood.