Don't Forget the Diver - S4-E2
Factual error: While Capt. Square is looking through his binoculars at what he thinks is Mainwaring's platoon marching with their rifles, a 1960's Bedford lorry drives past on the road behind them.
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.
Andy and Helen Get Married - S1-E1
Other mistake: Sam Jones' son, Mike, tells Aunt Bee that the farmhouse has been in their family for 150 years. While showing Aunt Bee a photo album, he points to a photo and says, "That's my great-great-grandfather who started this farm." Pointing to a woman in the same photo, Mike says, "That's my great-grandmother. She lives in Philadelphia." It is very unlikely that the daughter of a man who had started a farm 150 years earlier would still be alive. Assuming the man started the farm when he was in his 20s, he would have had to father a child when he was in his 70s and she would now have to be in her 100s.
Log 1: The Impossible Mission - S1-E1
Character mistake: They turn left onto Moorpark St and they say into the radio they are going east on Moorpark. However, the street sign shows they are going west. (00:10:30)
Episode #1.3 - S1-E3
Factual error: In the Scarlet Pimpernel sketch, the opening caption reads "France, 1778." This is the Scarlet Pimpernel who rescues people during the "Reign of Terror", as is confirmed several times during the sketch. But the French Revolution was still eleven years away and its Reign of Terror wouldn't start until 1793.