In the episode where Quinn becomes obsessed with what she believes is her guardian angel, she has a crisis of faith and asks Daria for advice. Daria takes the blunt tactic of simply stating that until proven otherwise, she doesn't believe in any afterlife. Something about the way Quinn says "But... that's so sad..." is just heartbreaking.
When Quinn's tutor rejected her and she began to cry in Daria's room. It was the first times she actually really liked someone, and for more than superficial reasons. For the first time in her life of guys blindly worshiping her, she gets rejected. It was one of her rare moments of genuine vulnerability to a guy.
"Boxing Daria", the entire episode. The sight of an over-sized refrigerator box causes Daria to recall parts of a childhood memory of her in bed, overhearing an argument between her parents and her dad storming out. This memory keeps nagging at Daria and leaves a real strain on her, until she finally remembers what the argument was about. Jake and Helen were fighting about her, due to the collective stress of working aggravating jobs and being called in by the school because Daria wouldn't socialize with the other kids. The box was important because Daria had one just like it when she was younger that was a sort-of haven. The entire episode has Daria on the defensive and very insecure. Seeing her be that emotionally vulnerable without bursting into tears and thinking that she's a cause of constant and unwarranted stress on her parents and family because she chooses not to socialize and "be herself" is almost painful. This is something that just about everyone has gone through in their life, and it's hard not to sympathize with Daria and be a little awed by her when you realize she began to reconsider the decisions she's made and wonder if she may have been wrong in making them the way she did.
In "I Loathe a Parade", Tom and Jane head off together, leaving a paint-splashed Daria behind. Tom steals a longing glance back at Daria, who fades into the distance looking utterly lost.
The end of "Dye! Dye! My Darling". After "The Kiss", Jane tells Daria that she and Tom broke up, and then they argue about the events leading up to that point. At the end, they break off their friendship (at least until "Is It Fall Yet?")
There's something incredibly tear jerking about Kevin and Brittany's implied separation in "Is It College Yet?", probably due to the fact that out of all the relationships in the show, theirs were always the closest and the happiest.
YMMV on that, but one feels sad for Kevin, who is left behind by his classmates in high school and now by his girlfriend.
In "See Jane Run", Jane joins the track team and becomes a big athletic star. Daria talking to herself is funny until it hits you that she's doing it to compensate for Jane being busy with practice all the time. Jane is her only real friend in the world, and not seeing her really takes its toll on Daria.
In "Is It Fall Yet?", Daria has finally made a bond with the outcast boy at the summer camp she is volunteering at. He starts to open up to her, so that she can help him... and O'Neil comes waltzing up and interrupts, making Daria's attempts at friendship sound more like manipulation. Incensed, Link storms off, followed by Daria, both looking like they want to cry at how everything went wrong in a matter of seconds.
In "Write Where it Hurts", Daria is frustrated about her inability to complete a writing assignment. Helen, after initially upsetting Daria due to her own stress problems, helps Daria to come up with a writing premise. Just hearing Helen speak so knowingly about Daria, which indicates that she cares far more than Daria realized, is touching. But the real kicker is the story itself: She writes about her family, years in the future, happy and well-adjusted. Her sister is raising a family, Daria's a professional writer, her father is much more relaxed and her mother is content. Underneath all the layers of cynicism and sarcasm, she loves her family and wants what's best for them all. Helen wasn't the only one tearing up after she read that.
In "Lane Miserables", Trent says he once lived in a tent in the yard for six months, waiting for someone to invite him back inside.
The outcome of Mr. O'Neill's effort to teach success through failure in "The F Word" was nothing but pure despair for almost everybody, especially for Brittany and Kevin. They may be idiots, but that doesn't mean you can help feeling bad for them.