When the first baboon arrives in Telepod #2 he hits against the glass door of the pod, leaving a bloody mark. In a subsequent shot the amount of blood has significantly increased and the shape of the mark is different too.
Tawny's panties change during her semi-nude scenes. When she was sitting on the sofa watching Brundle's teleportation, her panties are lacier. Later, when she was about to apply alcohol on Brundle's back, her panties have less lace trim.
When Seth brings Veronica to his lab for the first time, he is holding a padlock in his hand. We assume, since his experiments are secret, he locks the door when they are both inside. When she is getting ready to leave - after the tape recorder incident - the door is not locked when she goes to open it.
In the "steak experiment" sequence, Seth cuts the meat in two pieces, and teleports the bigger one. Later, Veronica eats a bit of this bigger half and finds that "it tastes like a steak", and when Seth gives her a piece of the smaller one and says "now try this teleported half", she finds that "it tastes synthetic". This is a continuity mistake. (It has been suggested that Seth deliberately misled Veronica as to which steak was teleported, to test Veronica's psychological reaction. However, this is unlikely; if the test were merely to determine Veronica's aversion to eating teleported meat, then it would not have led directly to the realization that the computer was incorrectly reassembling flesh).
When Veronica is playing the cassette tape of her recorded conversation with Brundel for her former boyfriend, the direction of the tape and the amount of tape left suggests that it just began from the beginning. However, the portion of the conversation that is being played is just before her tape recorder begins beeping, indicating that the tape is at the end and needs to be flipped over.
In the last scene when Veronica shoots the shotgun, in real life it would have ripped away from her, and probably injured her hands and wrists. You must brace a shotgun to the body to prevent injury from the recoil.
When Brundle is giving his 'gene pool speech' to Ronny, he's walking around shirtless in his apartment. In the long shot of him (eventually) walking out of the room, the hairy wound on his back is not present. Seconds earlier in the scene, it is there and Ronny comments on it.
When Seth is changing into a fly, he demonstrates how he can walk on the ceiling and walls like a fly. While upside down on the wall, he lifts his t-shirt to show the side of his body then places it back down. The reaction of the t-shirt to gravity (or lack thereof) made it obvious that he was filmed right side up and then the film rotated to get the effect.