Revealing mistake: When Rimmer is piloting Starbug in Lister's body, there is a shot of Starbug spinning in circles. This is a shot that was taken from S3 E2 Marooned, of Starbug crashing after colliding with the asteroid. The only difference is the planet in the background has been changed. (00:22:10)
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.
Continuity mistake: When Lister (in Rimmer's body), Cat and Kryten are chasing Rimmer in Blue Midget, the exterior shot shows all three sitting in one row. From the interior, Cat is sitting in front of Lister and Kryten, and Lister and Kryten are sitting behind Cat. (00:22:40)
Continuity mistake: When Rimmer (in Lister's body), is driving Starbug through the moon, the others are following in Blue Midget. One shot shows Blue Midget directly behind Starbug, but another shot shows Blue Midget much further to the right. (00:23:10)
Continuity mistake: One of Starbug's legs breaks off when it hits rocks on the moon but in the next shot, the leg has returned. (00:23:20)
Revealing mistake: During the chase on the moon, there is a shot from in front of Starbug crashing into an asteroid. About twenty seconds after this, the exact same shot of Starbug crashing into an asteroid is been repeated a second time, when the ship crashes. (00:23:20)
Revealing mistake: Starbug and Blue Midget wobble on a number of occasions during the chase. (00:23:25)
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.
Timeslides - S3-E5
Continuity mistake: During the golf game, Lister's locks move between shots throughout the scene. (00:01:20 - 00:02:20)

Timeslides - S3-E5
Continuity mistake: A blue fan appears on the golf course in the middle of the scene from nowhere. (00:01:30)

Timeslides - S3-E5
Continuity mistake: When Kryten notices the photo of the bike in the countryside is moving, the peg attached to it is near the middle, along the top, but in the shot from behind, the peg is much closer to the right. The same goes for the photo of Kryten's birthday party. (00:03:30)
Timeslides - S3-E5
Continuity mistake: When Kryten first sees the photos, in one camera shot they are moving on the string. In the next shot they are perfectly still. (00:03:40)

Timeslides - S3-E5
Continuity mistake: When the slideshow starts, the photos in Lister's hands disappear between camera cuts. (00:06:20)

Timeslides - S3-E5
Continuity mistake: When we see the photo of the skiers, there are no boxes in front of the screen. When the shot changes, two boxes have appeared which Rimmer and Lister use as steps. (00:07:20)

Timeslides - S3-E5
Continuity mistake: When we see the skiers, no one is standing in front of the screen. In the next shot, Lister has appeared from nowhere. (00:07:20)

Timeslides - S3-E5
Revealing mistake: When Lister is hit in the face with the snowball, some of the snow lands on Rimmer's hologramatic uniform. (00:07:40)
Timeslides - S3-E5
Deliberate mistake: The Eagle on Hitler's briefcase is that of the (post war) Federal Republic of (West) Germany - not that of the Nazi German Third Reich. This was deliberate - showing the Third Reich Nazi emblem of an eagle holding a swastika would have had the episode banned in Germany, Austria, Poland and any number of other European countries. (00:08:25)
Timeslides - S3-E5
Continuity mistake: When Lister is looking through Hitler's briefcase, he finds a banana and crisps sandwich with a bite taken from it. When Rimmer takes a bite out of the same sandwich later in the episode, the sandwich is whole. (00:08:30 - 00:26:15)
Timeslides - S3-E5
Character mistake: The assassination attempt on Hitler in 1944 was in Rastenburg, not Nuremberg as stated in this episode. (00:08:50)
Timeslides - S3-E5
Factual error: The newspaper detailing the bomb attack on Hitler lists the date as "Tuesday May 4 1938." May 4th, 1938 was a Wednesday. (00:09:15)