Community/Tear Jerker

In "Introduction to Film", the scene where Abed shows the video he made to his father. Relatively speaking.
The ending of Abed's Stop Motion Christmas episode counts, too, fully invoking the Narm Charm of the Rankin/Bass Productions Christmas Special.
Seemingly all Abed-focused plots. Even the Chicken episode was a bit depressing when you see Abed's obsession solely for his need to relate to people.
Pierce alone and unconscious on a park bench, having chosen his pill addiction over his friends, on Valentine's Day.
The Dean's breakdown in "Documentary Filmaking Redux". Turns out He didn't even go to Greendale, he went to an actual University and he felt like he was ashamed of Greendale, and wanted to make it better.
This one might be just me, but in the episode where Troy meets Levar Burton he cries a lot. Not out of happiness either, but because Pierce is tormenting him with what he most feared, disappointing his idol. It's Played for Laughs, but I didn't find it particularly funny, I just felt bad for him.
- Agreed. Maybe it's just because he's my favorite character, but a lot of Troy moments make me sad. Another one that stands out is in "Mixology Certification" when Troy has his first birthday celebration but Jeff and Britta ruin it by spending the whole night arguing. Troy was obviously able to shrug it off, but to me it was a serious bummer.
When ~~Omar~~ Prof. Kane is talking about Lego's, did anyone else find this really depressing? I mean, it's funny in a uncomfortable awkward kinda way, but still. It's a man fresh out of prison, he's been in their for god knows how long, and he's lost touch with society, and more importantly stuff he enjoyed before he got arrested. He knows everything changed but nobody understands what he's talking about, because they have been surrounded by it for the amount of time that he has been inside. It's depressing in a Fridge Sadness kinda way.
Pierce's home life in a fridge way. Before his mother died in the third ep of season 2 it was just him and her with maybe a few visits from his Jerkass father who passed down his prejudices. His seven marriages could be just his desperation to not be alone but for many reasons the marriages couldn't work, his "Super Fertility" is just a euphemism for being infertile (Which may relate to why he sees Jeff and Annie as his faux son and daughter), his friend Troy got to live in his mansion in season 2 for free before moving out (Since Troy's an idiot he had no idea that he was addicted to pills until it was too late) and didn't express much gratitude and from the season 2 finale that summer and most school breaks are just days to wait until the new semester.
Chang's Imagine Spot of him hanging out with the study group in "Accounting for Lawyers". He's in a Pop 'N Lock contest by himself because they were late and it feels like they abandoned him.
Early 21st Century Romanticism where Jeff realized the group didn't leave him any messages.
Britta's breakdown in "The Science of Illusion" where she tearfully reveals that she was behind the backfiring April Fool's prank that caused all the trouble in the episode because she just wanted to be wacky and fun like her friends for a change. Since they'd all been ragging on her for being a buzzkill (Jeff especially), they all look rather guilty. It then sets off a chain reaction of weepy confessions of feelings of inadequacy from everyone except Jeff and Abed, which are all quite sad -- until, of course, we get to Troy's.
- Even Troy's confession is a tear jerker in hindsight. He says that he only pretended to get the joke because he wanted Jeff to think he was smart. At this point in the series, Troy is constantly worried about what sort of impression he makes on Jeff - the cool, smart, grown-up lawyer. And we now know from Pillows and Blankets that Troy is "insecure about his level of intelligence." Insecure enough for Abed to consider it a weakness worth relating to his troops.
The ending of Contemporary Impressionists. You know you cried.
Troy and Abed completely turning on each other in Pillows and Blankets. Abed sends an email to his troops detailing Troy's weaknesses - including his insecurity about his intelligence, and the fact that it's incredibly easy to make him cry. Troy tells Abed that he'll never get another friend, because nobody will have the patience with him that Troy had.
- At the end of the episode, the two of them both want to keep fighting, because they're convinced it will be the last thing they ever do together. They would rather beat each other with pillows indefinitely than go their separate ways, but they don't see how they can just go back to being friends. Until Jeff intervenes.
Kind of a fridge tear jerker: Abed seems pretty upset (for Abed) in Regional Holiday Music, and he's desperate enough to spend Christmas with his friends that he resorts to recruiting them for the glee club pageant. It seems funny at first - and then you really look at him in certain scenes, like right before he and Mr. Rad start singing, and when he's apologizing to the rest of the group and saying he'll see them all after break, and he looks/sounds like he's as near to actually crying as he's ever gotten. And then you remember that last year, following a complete mental breakdown, Abed decided that the new meaning of Christmas for him is spending time with his friends.

Abed: We're inside a locker. It's where I spent a lot of time in junior high.
Annie: You think this is where we'd put you? You know that's absurd, right?
Abed: Well, I'm not stupid. You can see I've upped the square footage. It's a metaphorical locker. It's a place where people like me get put when everyone's finally fed up with us.

Professor Kane revealing that Starburns is dead, a reminder of what a skilled dramatic actor Michael K. Williams is.
- Even sadder when we learn that Kane blames himself for the death (likely because he knew about Starburn's meth lab and could have stopped it) and even quit in distress.
Gilbert Lawson discovering that his father never cared about him, and will only let him have the company if he refuses to ever tell anyone who his parents are.
- To elaborate, Lawson is black and his father is Peirce's more racist father Cornelius. So imagine spending your whole life mastering your father's game and winning and being presented with a legal document that makes it illegal for him to say who his real parents are. All because it would tarnish the pure Hawthorne name.
Britta. No, really. She is a very broken person. She tries to act tough but has everyone see through it and tear her down. She only dates guys who seem to treat her like garbage. All of this seems to stem from the fact that she was molested as a young girl and her father didn't believe her. Someone give this poor woman a hug. She needs it.
- Especially painful when you see evil timeline abed mentally break her down in the third season finale.
Sure, it was played for laughs most of the time, but Troy leaving the study group due to sacrificing himself to the Air Conditioning Repair School still feels very sad, particularly where Abed can't even face the study group anymore when he mentions that Troy told him as a parting word, "I know you hate this when people do it in movies."
Near the end of the Season 3 finale, when we see Abed destroying the Dreamatorium.
- But also sort of a fridge heartwarming, as it demonstrates that he realizes that perhaps he has some maturing to do. And he did keep a few panels of it, and built a smaller one.