Heart of Ice - S1-E3
Other mistake: When Batman is watching the security footage of Victor Fries' accident, at first Fries is addressing the camera directly which follows him and pans according to his movements and Ferris Boyle interrupting him, which would suggest a second person was in the room with a handheld camera. But after Ferris Boyle enters the room and orders Fries' experiment terminated, there are several cuts from various angles in the room suggesting surveillance cameras. The surveillance style footage also often conveniently cuts to the face of whichever person happens to be speaking or will cut to a certain angle in order to put emphasis on the action, all of which would require multiple mounted cameras and purposeful editing by either Boyle or one of his subordinates, which would be completely illogical since they don't want anyone to see the footage in the first place. (00:12:20 - 00:13:55)
The Tale of the Whispering Walls - S2-E7
Factual error: Betty Ann is supposed to be telling this story on a Leap Year night with a full moon. While it's necessary for the plot, the most recent Leap Year with a full moon was in 1972, and is calculated to happen in 2048 next.
Continuity mistake: In the episode where Fester invents a fabric: When Fester lies down on the parking lot, his head is first pointing upwards, towards a car, then in the next shot he has turned 90 degrees and his head is pointing towards Norman's legs.
Who Burned Mr. Brinker's Store?: Part 1 - S1-E6
Other mistake: The arson investigators name Jamal as the main suspect in setting fire to the store, because the security camera caught him sliding something through the front door mail-slot. Any arson expert would immediately know that it wasn't Jamal that started the fire, as the fire's point of origin was in the back of store, and was also entirely contained there, and Jamal was not only nowhere near the back of the store, but was also outside.