Plot hole: After the rock falls on King Tut at the end giving him amnesia, Batman explains how the rock fell. He says Tut raised his voice three decibels above high C. Decibels are a measure of volume, while pitch is a measure of frequency: the two are not directly related. This mistake was done in other episodes of Batman also, such as when Joan Collins played the villain.
Plot hole: When he abducts Alfred, believing him to be a minister, Penguin suffers a peculiar memory lapse. He's encountered Alfred twice before in previous episodes, even kidnapped him once before (knowing full well that he was Bruce Wayne's butler). Yet here, he fails to recognize Alfred at all.
Factual error: When Batgirl checks Batman's pulse, she states that his pulse is "three beats per minute." However, she only holds Batman's wrist for two seconds. There is no way she could have determined that pulse rate in so short a time. It is unlikely she could have detected even one heartbeat at that rate in two seconds.
Continuity mistake: After Batman has captured the last fruit fly, he goes to put the vial away in his utility belt. In the close up shot, his hands move toward the left side of the belt. The shot changes to a long view and his hands are now instantly coming up from the right side of his belt.
Factual error: Gotham City's museum should brush up on its paleontology. Their sign claims the neosaurus egg is 40 million years old. A) Neosaurus lived in the early Permian period, 290 million years ago. And B) All dinosaurs were rendered extinct 65 million years ago, most likely by an asteroid strike. So 40 million years ago, there were no dinosaurs on Earth.
Factual error: The British curator from whom Lord Ffogg steals a collection of artifacts needs to go back to school (or someone in Batman's prop dept. does). While most of the signs are correctly spelled and punctuated ("Duchess of Desborough's Diamonds," "Her Majesty's Priceless Snuffboxes," etc.), the one marked "Lady Easterlands Jeweled Easter Eggs" is missing its requisite possessive apostrophe. (The closed captioner noticed, though, and corrected the mistake there).
Continuity mistake: When the Batmobile pulls up to the theater where Tut's henchmen are emerging, Batman challenges them while standing at the open driver's side door. You can see his left hand closing the door, as well as hear it click shut - but in the next, reverse angle shot, as he and Robin leap into the fray, the car door is standing wide open again.
Continuity mistake: The Batmobile returns to the Batcave by remote control. It's facing left when we get a brief, three-second shot of Batgirl talking to Batman and Robin. In the very next shot, the Batmobile has instantly turned itself around and is facing right. Its turntable (which was actually pushed manually by stagehands under the platform) didn't work quite that fast.
Character mistake: Vertigris tells Joker he planted a bomb in the Batmobile while it was parked in front of City Hall/Police HQ. Only in this instance, it wasn't. It was parked in the alley out back, the better to hide his bomb-planting from public view, so it's strange that he'd forget this.