Revealing mistake: In the sleeping quarters, Rimmer turns on a swivel chair. This would be impossible as he is a hologram. 3.10. (00:03:10)
Suggested correction: As a hologram, this would also mean Rimmer could not touch or sit on the chair. The chair was likely a hologram as well, used to be operable by Rimmer. It has been shown before that Rimmer can still interact with other holographic objects.
Revealing mistake: When Lister is in jail with the Cat, as he is looking out of the window, he is naming famous people. He says "Mussolini" before he actually looks at him. (00:10:55)
Suggested correction: First, Lister is shown looking out the window throughout the scene, and second, he still has an entire view of outside the window within his eyesight. Lister likely already seen Mussolini within his eyesight before turning to face his direction, which he likely did to assure himself it was definitely Mussolini.
Plot hole: Lister and Kryten seem to know about Cassandra's prediction that Rimmer would sleep with Kochanski despite the fact that they were not there when Cassandra said it. (00:25:15)
Suggested correction: Rimmer told them, just not on screen.
Other mistake: The milkshake and toffee crisp come out of a different vending machine to where they were ordered. (00:01:50 - 00:06:55)
Suggested correction: But they are dispensed at the most convenient location for the person who ordered it. Great customer service, which can't be classed as a mistake.
Visible crew/equipment: When the Cat says "You mean to help you get out if the program freezes" on Blue Midget, a soundman can be seen crouching behind the control panel. The top of his head is visible in the top right of the screen. (00:14:10)
Suggested correction: That is not the head of a soundman crouching behind the control panel, or any crew member for that matter. What you are seeing is Kryten's hand moving around.

Other mistake: Rimmer's badge of rank is missing from his short sleeved uniform. All other uniforms in the series have the badge. (00:15:10)
Suggested correction: This is based on an assumption. This can't be classed as a mistake unless it's known as a fact that the short-sleeved uniform should also have the badge.
Continuity mistake: From the model shot of Blue Midget, Kryten is seen sitting on the right. When the camera cuts inside the ship, he is sitting on the left. (00:22:40)
Suggested correction: When the exterior shot of Blue Midget is shown, it is not very easy to tell the crew apart. They appear blurred, so it is impossible to determine who is sitting in which place.
Other mistake: When Lister inadvertently sets off the Auto Destruct sequence, the computer voice says the ship will detonate in 15 minutes. The computer then says "the ship will detonate" every five seconds followed by how much time until detonation, but the time in which the computer says this is inconsistent. Two seconds after the computer says 15 minutes, the computer says the ship will detonate in 14 minutes 55 seconds. Ten seconds after saying that, the computer says the ship will detonate in 14 minutes 50 seconds. (00:01:50)
Suggested correction: He keeps hitting it to cancel the self destruct, which then speeds up the countdown.
Plot hole: When Holly discusses what the crew can do as a result of activating the self-destruct, she comes up with three options which all end with "And get blown up." Since it turns out that Holly disconnected the bomb "ages ago," wouldn't she have known that they wouldn't be blown up?
Suggested correction: It's been a while since I've seen the show, but Holly is more than a little eccentric and silly, and if memory serves, it's implied he/she has gone a bit "senile"/glitchy due to sitting around mostly unused for three million years while Lister was in stasis. From the sound of it, it's likely just that Holly was being a bit thick and forgot in the moment about disarming the bomb.
Other mistake: The Cat Priest has no fangs when every other Felis Sapiens has them.
Suggested correction: How is this a mistake? It isn't known that every Felis sapien from 3 million years of evolution must have fangs.
Other mistake: Cassandra says that all the canaries will be dead in 1 hour, with the exception of Rimmer, Kryten, Lister, Cat and Kochanski. However, Kill Crazy was still knocked out in the transport vehicle and presumably would have survived too.
Suggested correction: But Cassandra was known to lie if it suited her purpose, she said that Lister would harpoon Arnold through the head and kill him but that's not what really happened either.
Yes, Cassandra lied about Lister shooting Arnold through the head with a harpoon, but that was an attempt to stop Lister from killing her. Mentioning which of the canaries will survive had no impact on any of her plans, so not mentioning Kill Crazy survived is still incorrect on her behalf.
Kill Crazy was safe in the transport vehicle, so wouldn't be affected by the station flooding. Mentioning him would have been irrelevant.
Other mistake: Throughout the show, reference is made to Rimmer's light-bee, a solid unit that buzzes around projecting his holographic image. Yet he constantly walks through things without any difficulty. Not physically possible if there is something solid in his form.
Suggested correction: Rimmer seems to have been projected externally in the first two series and switched to the internal light bee before the start of series 3 (maybe Kryten brought it over from the Nova 5). This would explain how he was able to pass through solid objects in the first two series and how his legs managed to go off on their own in "Queeg." Although from season 3 onwards, it shouldn't be possible.
Other mistake: When the group are discussing if they can override the Auto Destruct, the computer voice says 8 minutes 20 seconds until detonation. About 20 seconds after saying this, the computer voice then says 8 minutes 10 seconds until detonation. (00:02:35)
Suggested correction: This isn't a mistake; the system is malfunctioning. Getting the time wrong is very inappropriate.
Revealing mistake: After Starbug has left Red Dwarf, the scene cuts to inside the ship where Rimmer tells Kryten to autopilot, and the background of space is entirely still, even though the ship is moving. (00:04:45)
Suggested correction: Stars are so far away that they do not appear to move when travelling in space, unless you reach incredibly high speeds or change direction.
Other mistake: When the Cat says there better be no dogs on the ship, there are only about two or three stars visible out the window. Far too few given the amount of stars shown in exterior space shots of the ship. (00:05:15)
Suggested correction: They aren't stars; it's dust on the window... (a reference to a later episode).