Deliberate mistake: Robert Brown wears a gold turtleneck sweater with his leather outfit throughout the series. Though it's more than 100 years beyond the show's 19th Century setting, he and the producers decided that he looked good in it, so allowed the anachronism to stand.
Log 1: The Impossible Mission - S1-E1
Deliberate mistake: Halfway through the episode, they are chasing car thieves in a drainage channel. The thief's car comes to the end of the road and crashes into a deeper gully. You can see that the windshield has been taken out of the car, although it was previously there. They cut away and when they come back, parts of the windshield are back in the car, but the whole center is missing.