Continuity mistake: When V'Ger first appears on the bridge as a column of light, you can see that the two sides of the image don't match up, because they tried to splice the special effects person who was moving the column out of the shot.
Continuity mistake: In the 2000 Director's Edition only: during the scene in which Kirk and McCoy are asking Spock why he has returned to the Enterprise, they added an effects shot of the Enterprise's warp nacelle outside the window. Keep an eye on it throughout the scene. Even considering the fact that the camera angle changes throughout the scene, the nacelle appears to "move around" (side-to-side, up-and-down, nearer/closer to the window) and the perspective/foreshortening of it never changes in relation to the camera angle. (00:53:35 - 00:56:30)
Continuity mistake: Near the end of the movie, Kirk reads the nameplate & begins to wipe off the "V-GER" sign, he uses his bare fingers to clean the sign. As he circles "V-GER" you see his hands beside them and they are perfectly clean. His hands should have shown some evidence of wiping off the VOYAGER sign.
Continuity mistake: After Spock's arrival, the Enterprise is finally able to exceed Warp 1. At the end of the test run, the shot shows the Enterprise, the beautiful light spectrum tube, and the pop/explosion created by warp speed. The problem is the pop/explosion and the spectrum tube do not line up. It's almost as if the two started at different points.
Continuity mistake: After Ilia is abducted by the V'Ger plasma energy beam on the bridge, her tricorder falls on the chair. The chair is facing a different angle than it was before. Also, you can see the white tips of a pair of shoes from someone standing next to the chair on the left. Presumably, this is supposed to be Kirk, but he is too far away to see his feet in that shot.