Though this only shows up in the first episode, Plasmus is actually a normal human that becomes a monster whenever he wakes up. That means he only has two choices: sleep his entire life away, or become a dangerous threat to everything around him. He know this too, and he reacts in horror when Slade wakes him up, which seems to indicate he's a fairly decent guy in a really sucky situation.
This entire exchange at the end of "Masks" was deeply saddening, considering Robin and Starfire's relationship dynamic, Starfire's values on friendship and her usual bubbly personality, and what Robin had just done. Ouch.
Robin: You want to yell at me too? Everyone else has. And I don't blame them.
Starfire: I do not wish to yell, merely to understand. Robin, why did you not tell us the truth?
Robin: I needed to fool Slade. If you guys knew it was me under that mask, you would've held back. Doesn't matter anyway. Slade figured it out, and I still haven't figured out anything about him.
Starfire: That is not true. Whoever Slade is, you and he are similar. Slade did not trust you...and you did not trust us.
The schoolgirl's/Terra's speech. "Things were never the way you remember..." "Things change, Beast Boy. The girl you want me to be is just a memory.... you're the Teen Titan. That's who you are. That's not me. I'm not a hero. I'm not out to save the world. I'm just a girl with a geometry test next period and I haven't studied."
The end of "Aftershock, Part 2". 'Nuff said.
Beast Boy:Her name...was Terra. She was gifted with tremendous power, and cursed with it as well. She was a dangerous enemy, and a good friend. And she was one of the bravest people I have ever known.
During BB and Cyborg's battle with Terra. Beast Boy's expression of disbelief, and her utter lack of remorse make it heart wrenching.
Beast Boy:Terra, you can't.
The end of "The End, part 1", particularly as Raven accepts her destiny as herald of the apocalypse, then stuns her friends into unconsciousness to keep them from going after her...then whispers, "goodbye... be safe." Similarly, a few minutes later when she says she never knew she would "have such wonderful friends," and proceeds to "die" as her horrified teammates watch, helplessly.
Robin - normally not the type to give speeches about The Power of Friendship - doing his best to talk Raven into letting the Titans help, as well as Robin protecting Raven's young form and (for a happy-tears variety) the two Titans watching the sun come up at finale's end.
"The End: Part 3", when Robin was carrying regressed-to-childhood Raven and telling her the story of the Raven he knew. One of the sweetest moments in the show. And when young Raven was shaking Robin's body, going: "Get up, get up!"
The Titans fighting for their lives against there evil doppelgangers. The originals keep trying to fight the dark selves, but keep getting beat down.
Doppelganger Beast Boy: (to Beast Boy) No wonder Terra dumped you.
A-few-punches-from-death Robin collapsing into Starfire's arms, whispering, "It's okay, Starfire... s'okay...".
In the same episode, when Starfire was forced to hit Robin (who was Not Himself at the time) unconscious with a starbolt. The resignation on her face... damn.
From the same episode, it's also quite heartbreaking when Beast Boy insults Raven after she refused to play Stankball with him and Cyborg. The look on Raven's face is particularly sad. Anyone who knows what it's like to get insulted by their friends for being different definitely felt for Raven there.
Beast Boy: Why can't you just have fun like normal people? Why are you always locked in your dark room reading nasty old books? Why do you have to be so creepy?
Raven: (very quietly, to herself) I'm not creepy, I'm just different.
Also Raven, of all people, crying over how lonely she thinks she is. Ouch.
Melchior: Is that what you want, Raven? To be alone?
Raven: ... no.
Worse still because he is emotionally manipulating the hell out of her.
In Stranded, there's a sad moment when the Titans are trying to shake off an alien on their T-Ship, and Robin tells them to not separate, but the alien's sound waves mess up the communications, and the Titans think they're told to separate, and do so, as Robin extends his hand out and lets out an anguished Big "NO!" as Starfire's ship leaves.
In the same episode, Starfire expresses her disappointment in Robin for pulling the She's Not My Girlfriend card and worries that they'll never be okay as she puts it.
Her cold demeanor towards Robin while their alone on the planet also counts too all because Robin Can't Spit It Out.
In the final battle from, "Cyborg The Barbarian" the Big Bad Krall has Sarasim in his grip and looks about ready to kill her even as Cyborg is being carried away by his army. Cyborg is diverting all of his remaining power for one final shot...only to be torn from the field of battle and brought back to the present by Raven. When he demands that they send him back to finish the battle, Raven tells him that she can't.
Cyborg: I made a promise! They were counting on me!
Robin: Cyborg, I'm sorry. We had no idea...
Cyborg: I didn't even get to say goodbye.
Sarasim's reaction to seeing Cyborg trying to save her counts as well. Throughout the episode, Cyborg depletes more and more of his power and suffers damage he cannot fix in her time. She knows that the energy he uses for his sonic cannon is also the energy that keeps him alive, and how little energy he has left. She knows that he's willing to sacrifice his life for hers, and there's nothing she can do to stop him.
From the movie: the Fate Worse than Death that the Commander of the Tokyo Troopers subjected Brushogun to after capturing him was so cruel and painful to watch.
Cyborg's origins, while never discussed, do make you feel bad for him. It's implied in "Deception" that he wasn't always a cyborg, which is pretty sad. You can tell by looking at him that he's glad to have his human body back. This only adds onto the tragedy of his story, whatever it may be.