This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / American Horror Story: Asylum
Lana learning that she has been committed by her girlfriend.
Lana discovering that her girlfriend has been murdered by Bloody Face/Dr. Thredson, and that she is now his prisoner and they will "continue" her aversion therapy. By using her girlfriend's corpse.
Lana's final confrontation with Dr. Thredson. What really makes it a tearjerker is that Lana is only confronting him because she wants to know what Bloody Face did with Wendy's remains. Lana comments how she's the only person who cares because Wendy was a lesbian. Somewhat sad Truth in Television.
YMMV on this one, but the look on Shelley's face whenever the schoolchildren run away screaming from her. She just wants someone to help her out!
The way she finally dies is even worse.
Dr. Thredson's origin story, but that quickly crosses over into Nausea Fuel and Squick when he mentions the medical school cadaver.
The Alas, Poor Villain death of Dr. Arden. Heavy YMMV due to him being a Nazi, but here was something who yearned to be considered a 'visionary' and was so twisted that he was unable to form anything approaching a normal relationship. He laments that he never had any innocence, even as a boy, and reveals that he is exceptionally lonely...the only solace he takes is in his own intelligence and the innocence of Mary-Eunice. However, Pepper informs him that the aliens (who he sees as equals) laugh at his experiments and consider him a quack, meaning the 'path' he followed was all for nothing. He loses faith in his own motivations, ending his experiment while declaring that his life's work is a 'farce'. Then Mary-Eunice dies, leaving him completely alone in the world. Distraught, he tearfully burns himself alive with her. Deserved due to his horrible acts, but somewhat sad nonetheless.
Everything the Monsignor goes through.
"Take me away."
Sister Jude/Judy reacting to Pepper's offscreen death.
Sister Jude's death.
Johnny and Lana's confrontation, ending in his death.
90% percent of all characters in that picture on the main page? They're all dead.
Pretty much any scene involving Shachath is guaranteed to be a tear jerker.
Sister Jude more than earned a comeuppance, but her Trauma Conga Line is just awful. When her sanity finally breaks in The Name Game, the sudden cheerful, happy song just makes it all the worse.
Mother Superior Claudia, the only character on the show who's never wished harm on anyone nor asked for anything for herself, sacrifices her career to free Lana from Briarcliff. It's implied Cardinal Howard reassigns her out of revenge, but nobody — Lana and Jude included — cares to follow up on her.