From A Very Potter Musical

"Missing You". Harry's sense of loss over Dumbledore's death and Quirrel's longing for Voldemort are both pretty gut-wrenching, even when the lyrics are humorous.
Harry: ♫ You gave me hope when my spells weren't right. ♫
Quirrell: ♫ You gave me someone to hold every night... ♫
Ron, Hermione, and Ginny's reactions to Harry announcing that he is going to sacrifice himself and Harry's good-bye.

From A Very Potter Sequel

Strangely, Dumbledore's rejection of Umbridge and her subsequent breakdown is quite depressing... at least, until Umbridge subsequently crosses the Moral Event Horizon.
Draco's letter to his father. The Parental Neglect from Lucius is rather sad.
"Guys Like Potter" will make you want to hug Snape, no question.
Soon after "Those Voices," Snape looks into the Mirror of Erised and sees Lily. He looks at her for a second, and then silently exits the stage in order to protect Harry and the others. Lily is left alone, and the lights fade to black...
"To Have a Home", which resonates with anyone who's ever been rejected then found a place where they truly belong, and Guys Like Potter, which, despite its last line, deals with the always-tearjerking Snape/Lily ship in a way comparable to the Prince's Tale chapter.

"I Was" is quite depressing, capturing both Harry and Tom Riddle's sadness regarding being unwanted and Harry's nostalgia for his old adventures.
"Everything Ends". The title alone kind of explains why.
"It's all that I loved, and all that you'll see..."
"OK is wonderful."
"Things come and go. Nothing lasts forever, kid." and "You know, there's two things in this world I love. One of them's you." Fucking hell.
Quirell and Voldemort's speech about Harry Potter, how it's brought them joy, family and love but sometimes you need to let things go.
"Harry, Hermione's Petrified, and I can't do anything to help her!" Joey Richter's delivery is incredible.
Several of the actors are visibly tearing up during the last couple scenes. Several people have stated Lauren Lopez in particular got them crying too.
All of the other characters lining up to hug Harry goodbye, especially as most of the actors are genuinely crying. Special mention goes to Neville's goodbye—while the other farewells are softened by the overall ridiculousness of each character, Neville just gives a quiet, understated, "Goodbye, Mr. Potter," and presses his face into Harry's shoulder during their hug. The whole thing is clearly just as emotional for the actors as it is for the characters.