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)
Other mistake: Nefertiri has been buried for almost 2000 years, yet speaks perfect modern English.
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.