Deliberate mistake: 9-12 "The Bells Toll at Noon": When McGarrett picks up Danny and drives away in his Mercury Grand Marquis, the rear view mirror is missing, probably removed to prevent it from blocking the actors' faces. It reappears later on. (00:33:00)
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.
Deliberate mistake: We see the Air Force plane's bomb bay doors come open. But in the next shot of the plane banking, the doors are instantly closed again. (00:45:20)