While it's mostly played for laughs, Lister learning everyone is dead after he first comes out of stasis in "The End" was pretty sad to watch. Especially when he realizes Kochanski is gone.
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.
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 argument Rimmer has with himself in "Me2", where he uses his parents abuse and neglect against himself.
Rimmer: Mummy did like me, she was just busy!
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 realizes. 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.
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 just my song."
The song in question is "Someone To Watch Over Me" by George and Ira Gershwin.
Not to mention the line before the song:
Rimmer: ...My swimming certificates, my telescope, my shoe trees...I'd trade everything in to be in love, and to have been loved
Rimmer's rejection of the love Lise showed him, purely because it wasn't his memory to begin with. Especially as Lister claims Rimmer ultimately felt more for Lise than he ever did - whether or not Lister was telling the truth is never spoken of again, but it sure looked that way.
Rimmer: The woman I loved most in the whole world, didn't love me. She loved you.
The moment when Rimmer says he was mad to have given Lisa up, and Lister agrees, having come to the same conclusion about himself.
Holly in "Queeg" just before his deletion. Holly gives the crew all very nice farewells, then he sings the song "Goodbye to Love" as his voice fades out, and we see the words "ERASE COMPLETE" on the monitor...
Holly: Well, dudes, I've come to say goodbye.
Lister: So you're definitely going to get rubbed, Holly?
Holly: 'Fraid so.
Rimmer: Life's going to be hell!
Holly: Well, see you, Dave. Hope it works out with you and Kochanski.
Lister: Cheers, Hol.
Holly: See you, Cat. Hope one day in the not too distant future you fufill your heart's desire and get your end away.
Cat: Thanks, man.
Holly: And Arnold — Well, I hope you meet those aliens you're looking for, who can give you a body, and you become an officer and you get a sex life, and all the other millions of things you feel you need to make you happy.
Rimmer: Thanks, Holly.
Holly: Well, I hate long goodbyes. Perhaps next time you've got the dosh together to go down the disco, you'll raise a glass to your old mate Holly, and think "Things weren't too bad when he was around. Perhaps not the most efficient computer ever invented, but we had a giggle." Oh, one last thing — 45.265881
Holly: That's the square root of two thousand and forty-nine. I may not be fast, but I get there in the end.
Subverted moments later when it's revealed that Queeg IS Holly and has been this whole time. The entire episode, Holly was playing an elaborate joke on the crew because he felt underappreciated.
Lister playing his guitar one last time before he burns it in order to stay warm in "Marooned". Sure he still can't play, but that tone was just heartbreaking, despite the fact he decided not to go through with it a second later.
Finding out the trunk Lister destroyed was the only thing Rimmer's father ever gave him, the only hint of any sort of affection.
Lister's despondency upon learning Kryten is nearing the end of his service life in "The Last Day". Even Rimmer finds it sad.
In "Justice" when the prison A.I. scans Rimmer for guilt, and it picks up on his feelings of remorse over the accident that got the crew killed three million years ago. Sure the subject is normally played for laughs, with Rimmer all Never My Fault. But it's in this episode that we learn for the first time just how much the accident really haunts him.
Rimmer and Ace throughout "Dimension Jump", Ace tries his damned hardest to be nice to Rimmer. And as always, Rimmer just goes around making baseless accusations and barks insults.
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.
Lister then swallows Rimmer's light bee like last time, but it was messing with Rimmer's light bee that caused him to go nuts (well, more than usual) in the first place...nice job breaking it Lister.
The off screen execution of Winnie-the-Pooh, even if it was played for laughs. The look of shock on Lister's face shows how much it has traumatised him.
Lister: That's something no one should ever have to see.
In "Holoship" when Rimmer confesses to Nirvanah he cheated on his test and explains why.
Rimmer: Every time I look in the mirror, I see this. (points at his letter H) Only to me, it doesn't mean hologram. It means half-wit, hopeless, hideous failure. This was a chance to be somebody. Somebody I liked.
The shocked look on Rimmer's face when he learned Nirvanah was his opponent, and had sacrificed her place (and her existence).
Watching the alt!Rimmer and alt!Cat being vaporized by The Inquisitor near the end of his title episode.
The alternate Lister and Kryten seem to be pretty decent guys, before they get killed horribly by the Inquisitor.
Watching Lister in "Demons and Angels" being forced to murder High!Lister and High!Rimmer while under the control of the Lows.
While they were naïve to the point of idiocy, watching all the Highs being murdered by the Lows wasn't quite as funny as the laughter of the studio audience would lead one to believe.
High!Rimmer's demise seems the most tragic, as he at least appears to appreciate what's happening to him.
Even though it's restored at the end of the episode, the sight of Red Dwarf itself exploding. Made worse by the fact that they blew up the original model for that shot.
Near the end of "Back to Reality", before it's discovered they're only hallucinating Lister learns that in real life he's a mass murdering fascist. Kryten then shoots and kills a man, which obviously traumatizes him.
Near the end of "Out of Time", after the crew refuses to help their future selves upon seeing what they have become. The future crew out right viciously attack their past-selves, actually willing to kill their younger selves. When Rimmer shows uncharacteristic bravery, "Better dead than smeg!" we're led to believe we'll see a Moment of Awesome. Instead were treated to watching Lister, Cat, and Kryten all dying horribly.
Ace: Well done, Arnie; you've done us proud. Smoke me a kipper. I'll be back for... (Ace's Light Bee then crackles and his image glows white before fading completely)
And Rimmer's reaction, he's just so shocked, the closest thing he's had to family in over three million years blinks out of existence right in front of him. And not just family, the closest thing his had to family who's ever been kind to him.
Cat: What's happened to goalpost head?
Rimmer: (still in shock) No, you don't understand. It's not me, it's him.
The theme that plays when Lister and Rimmer discover the space cemetery. There's so much depth there, the music encapsulates the beauty of the scene.
Lister leaving his son (himself) under the pool table in "Ouroboros" and stating that he's never going to be alone.
"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 loss of Starbug in the opening of "Back in the Red", even if it was a rundown rust bucket held together with nothing but mold. It was home for the crew for years, and unlike "Out of Time", it's gone for good.
While sucking up to Captain Hollister, Rimmer mentions it's Hollister's anniversary and that he must be missing his wife terribly. Now take into account the fact that Mrs. Hollister's been dead for over three million years...
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.
Back To Earth
Interestingly enough, several of Lister's scenes in "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 meeting Kochanski in the hallucination. You can see how heartbroken he is knowing that she isn't real.
Kochanski: In your dreams, did you kiss me?
Kochanski: Did you enjoy it?
Lister: I never wanted to wake up.
Kochanski: Then don't.
Lister's plotline in "Dear Dave" consists of him trying to simultaneously cope with still being the Last of His Kind and the possibility that one of his old girlfriends had his child.
The flashback seen in "The Beginning" during Rimmer's time as a student at college on Io. His own father is the teacher (naturally). Rimmer's father uses his own son as the unwitting guinea pig for a class project about psychology. Rimmer's father then makes the students ostracize Arnold so he'll feel unsure of his decisions. ...Smeghead.
Considering what we learn in "The Beginning", Mr. Rimmer had been very aware from the beginning that Arnold wasn't his son. So it's highly possible all the abuse that Mr. Rimmer laid at Rimmer's feet growing up was simply doing everything he could to spite his wife and "Dungo".
In "M-Corp" Lister is implanted with a perception filter that makes him unable to see anything that isn't an M-Corp product. This includes Rimmer, Kryten and Cat. There's a scene where he goes through his morning routine fumbling around for objects he can't see (his toothbrush, the toaster, a breakfast plate, etc.). Could easily be played for laughs, but it's accompanied by rather somber music that accentuates just how LONELY Lister's existence is now, as if it wasn't already.
Infinity Welcomes Careful Drivers has an extended version of Lister's reaction to learning he's the last human alive anywhere, stuck on a giant red trashcan three million years from space, all as an end result of his attempt to just try and get home. Namely, a complete mental breakdown.
Kryten knew that the three female officers of the Nova-5 had been dead for centuries. He just chose to ignore that fact so he could keep himself from being forced to shut himself down. When the Dwarfers force him to confront the fact he's been caring for Skeletons all he can do is admit he was lying to himself and start the process that would essentially kill him.
Rimmer's twoHeroic Sacrifices at the end of Last Human. He finally overcomes his many neuroses and wins the approval of his son by taking on Lister's other self and to defeat The Rage. Kochanski cries, knowing what his final messages entail, Lister realises what Rimmer's small-minded bureaucratic tendencies were all about and even The Cat is shocked at his death.