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Tear Jerker: Red Dwarf
While it's mostly played for laughs, Lister learning everyone is dead after he first comes out of stasis in the first episode was pretty sad to watch. Especially when he realizes Kochanski is gone.
Interestingly enough, several of Lister's scenes in "Red Dwarf: Back to Earth" can potentially qualify for this. Watching Lister break into tears as he attempts to read Sense and Sensibility to Kochanski's grave shows that Doug Naylor can write excellently, a stark contrast (in the eyes of some) to the last two series of the show. It's only made more powerful by the fantastic music and the stunning (considering the shoestring budget) setting in an observation dome.
Lister leaving his son (himself) under the pool table in "Ouroboros" and stating that he's never going to be alone.
The flashback in "Balance of Power" of Lister having fun with his friends, annoying Rimmer and being teased about Kochanski.
Past!Lister: All right guys, leave it alone, leave it alone, leave it...
*Fade to the present, where a forlorn-looking Lister, the last human being alive, is sitting by himself in the empty bar*
Lister: ... alone.
More or less anything explaining why Rimmer is the way he is is unbearably tragic.
Future!Kennedy's last words in particular.
Thank you for giving me a chance to... be reborn.
"Epideme", at the part where Lister decides to kill himself and the virus with him rather than allow the rest of the crew to become infected
The scene in "Better Than Life", where all that post comes through from the past. Rimmer gets one about his dad being dead, Lister failing to figure out the scrawling writing for ages, then finally Rimmer realises. He's been dead for millions of years but it sort of just hits him here. The next scene shows Rimmer on the observatory tower, just musing to himself. Lister comes up and actually asks if he's okay, and Rimmer goes on about how while he hated his parents, he just wished he had been able to impress them just once. The end is ruined by the Cat making a joke, but still, very well written stuff.
Watching the alt-Rimmer and alt-Cat being vaporized by the Inquisitor near the end of his title episode.
The seventh-series departure of Rimmer when he takes up Ace's mantle.
The ending of Series VIII ("Only the Good") might also fit the bill. Rimmer returns from the mirror universe (he actually comes ''back'') to find that the ship is burning, the others have escaped and he's stuck. Then he discovers that the antidote formula he brought back is actually the formula for the virus. And the paper burns up. With violin music. It's the quiet little "smeg" as he collapses.
Watching the ship itself falling apart.
The oft overlooked episode "Thanks for the Memory". The scene where Rimmer, drunk to the gills, is talking about his regrets is actually quite touching.
Rimmer: "That was going to be our song... but I never found anyone to share it with. So now it's my song."
Also from that episode Rimmer's rejection of the love Lister showed him, purely because it wasn't his memory to begin with. Especially as Lister claims Rimmer ultimately felt more for Lisa than he ever did - whether or not Lister was telling the truth is never spoken of again, but it sure looked that way.
The end of the episode "Waiting For God" shows an old cat priest who's believed in Cloister (Actually Lister) his entire life. He then rants about how it was all a lie, lamenting all the time he wasted worshiping his god. It's made worse by Cat's complete disregard for the old man's speech. Crosses into Heartwarming when Lister does show up, though.
The ending of "Meltdown", when it's revealed that Rimmer's "tactics" ended up wiping out the wax droids entirely. Lister clearly viewed the droids as being sentient, and he's distraught to hear about how they're all gone. The fact that Rimmer is oblivious to why Lister would feel that way only makes it worse.
Rimmer: I brought about peace. Peace, freedom and democracy.
Lister: Yeah, Rimmer. Right. Absolutely. Now all the corpses that litter that battlefield can just lie there safe under the knowledge that they snuffed it under a flag of peace and can now happily decompose in a land of freedom. Ya smeg head.