Yuta Asahina being blown up during an attempt to escape the city and get help for Komaru and Toko. Komaru falls to her knees and contemplates giving up right then and there. Toko is also quite affected the death, and one of the more poignant parts is that for all she dislikes Aoi, Toko dreads having to tell her about her brother's death, possibly considering how she took Sakura's suicide in the original game.
Right before this happens Toko slips up and calls Yuuta Swimming Idiot, the nickname she always used for his sister. Really shows just how she projected her image on him, making her reaction to his demise that much worse.
Taichi's death. Especially how his last words are the expression that he'll be able to see his son again. The fact that you find a photo of him and Chihiro afterwards in his wallet is especially painful.
Even worse, when Komaru finds said photo, she says she hopes that Taichi's 'daughter' is safe. Toko turns away with a pretty conflicted expression, clearly not wanting to tell her the truth.
Any of the Warriors of Hope's breakdowns. Especially Nagisa. Poor, poor kid.
Masaru's breakdown is extremely hard to watch. This boy started out bragging about how he was going to win, before quickly devolving into a hysterical sobbing about how he refuses to be afraid of the pain and the "alcohol smell" anymore, and that he vows to protect all children from that fear because he's the "hero." Masaru's left arm starts shaking due to his rant, so he proceeds to beat it with his other fist until its bruised and motionless, screaming at it to stop moving and to listen to him. It gets even worse when you realize he was the victim of abuse by his alcoholic father, and hitting his arm is a callback to what his father used to do to him. Ultimately, in his mind his actions are to protect other children from going through what he did, no wonder he calls himself a hero.
This is somewhat mitigated by the fact we see a Holocaust style body pile-up that Masaru was responsible for not even a minute prior. The mood whiplash is so abrupt that it might be difficult to feel sorry for him.
Chapter 3's focus on Kotoko was not only notorious in giving Despair Girls its reputation for being Darker and Edgier, but also it being highly upsetting to watch. Watching a child relive memories of their sexual abuse is horrible and disturbing enough on its own, but Kotoko begins breaking down from the beginning of the chapter, unlike Masaru and Jatarou. This is brought on by Monaca calling Kotoko gentle, which basically reduces Kotoko into terrified and crying mess, begging to not be treated 'gently'. Both voice actresses for Kotoko, English and Japanese do a heartbreaking job at delivering Kotoko's lines, however Kazusa Aranami's voiceover is particularly upsetting as she eventually becomes hysterical while sobbing.
The worst part? These kids, convinced they had no other way out, were all planning to kill themselves by jumping off the roof of Hope's Peak Academy. Well, four of them were. Monaca just thought it would be funny to watch.
In Chapter 2, there's a note left behind by a man who's apparently about to be killed by his own son. He wonders if his child feels anything for his parents, and desperately hopes that it's a dream, but will accept being killed at his son's hands. It drives home how terrible the idea of children killing their own parents is.
Fridge Horror makes this much worse, since it's stated many of the children wearing Monokuma helmets are brainwashed. Not only are kids killing their parents, but most aren't doing it willingly.
Toko's discussions about her childhood are pretty sad at times. At one point she admits she's afraid of the dark because what's implied to be her parents locked her in a dark closet for several days. Not to mention her Kill List character is a stink bug.
In Chapter 4, you find a memo written by Nagisa where he states that he actually felt guilt over his actions, something he never told his friends, not even Monaca. In the end, the only reason he's going through is because he wants to create a world where children don't have to live in fear and for Monaca's sake, closing off by indirectly stating he loves her. Given the surrounding events, it just drives home what kind of person he was in the end.
Shirokuma's Heroic Sacrifice in Chapter 4 can be pretty heartbreaking. After all, he was the only non-adult, non-human friend you had. However, this is subverted after the revelation that he was part of Junko's AI.
Komaru's breakdown near the end of Chapter 5. Monaca shows her the bodies of her parents, sending her over the edge. She doesn't care about what happens to the adults, or the children, or the town, or the entire world anymore. When she tells Toko to shut up, and then says that she 'doesn't give a damn what happens anymore' (the second time she swears in the entire game, first to her motivation speech to the adults) is absolutely heart-wrenching.