Big-Lipped Alligator Moment: The scenes of Judy's mother (and her boyfriend) come out of nowhere and clash in tone with the rest of the film.
It makes sense when you learn that the movie was originally intended as a straight-faced B-Movie parody and this scene was a remnant of the original product.
There's also a scene early in the film where Ted Nelson and his wife argue interminably about crackers. Mike and the Bots are completely baffled, comparing it to the steak scene from Raging Bull; they make it into a Running Gag.
Critical Research Failure: It takes about six-and-half-years to get from Earth to Saturn. How did Melty get back to Earth so fast?
The novelization has Steve on a landed Mars expedition, so the travel time is far less (though still a long trip.)
Darkness-Induced Audience Apathy: The protagonist is a loser, the villain is a pathetic Woobie, Destroyer of Worlds, the violence and gore quotient is through the roof, and virtually everyone with a speaking part dies, including the hero. Then it ends with the villain melting and his remains scooped into a bucket. Certainly among the most unremittingly bleak movies to grace Mystery Science Theater 3000 (and yet is one of the funniest as a result.)
Designated Hero: Ted Nelson doesn't do a single heroic thing at all. Hell, the woman who cut off one of Melting Man's arms when he tries to kill her comes off as more heroic.
Narm: Plenty; the nurse's slowed-down running scene near the beginning, or the fisherman's line, "Don't scare away the fishes."
The elderly citrus thieves are completely bonkers.
"DON'T SHOOT! I'M TED NELSON!" (blam!)
When Steve rips the bandages off his face and sees that its starting to melt, he goes berserk and starts trashing the hospital room... with a strip of bandage still sticking to his nose.
Mike: Uh melt, you got a little something there.
All of these are almost certainly artifacts from the original plan to make the film a spoof. There are tons of things left in the flick that could have very easily been played for laughs, only they weren't.
Narm Charm: Helen and Harold, the elderly couple, are funny and adorable.
Nausea Fuel: Pretty much the entire movie. You wouldn't have wanted to be the person who had to mop up the theaters where this was screened. The astronaut even leaves puddles of himself behind with every step!
The Melting Man's death scene is particularly gruesome, though. (But hilarious!)
Padding: Dear God, yes. The same "You've never seen anything til you've seen the Sun through the rings of Saturn!" transmission, and so on.
They Wasted a Perfectly Good Plot: The idea of a man exposed to outer space radiation and his physical appearance and overall sanity slowly deteriorating as a result could've made a good horror movie if not for the shoddy writing, dull characterization, and loads of Artistic License.
Visual Effects of Awesome: As noted above, Baker's makeup effects. While not enough to save the movie, the effects are widely agreed to be one of its few saving graces.
Wangst: Okay, we get it - more horror movies have a callous attitude towards death and that death itself is not something that should be taken lightly, but everyone in this film, especially Ted, wildly overreact to it.