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.
Other mistake: While Batman and Robin are ascending in the balloon, Robin picks up a chewed piece of gum, and stops the altimeter just before 20,000 feet. After that, Robin wondered how to get down. Beside Batman's head is the handle of the cord that would have started to let the air out, and started dropping them down.
Continuity mistake: After the Mad Hatter's spray turns Batman's cowl pink, Batman and Robin are shown in the Batmobile, headed back to the Batcave. The stock shot shows Batman wearing his regular blue cowl while inside the car. Once they get to the Batcave, Batman's cowl is pink again.
Continuity mistake: When Batman is strapped into the electric chair, his utility belt is the standard flat-front, blank, yellow boxes. Once Batman breaks the wire loose to send a message, there is a large, black and yellow button attached to the the right front of the belt.
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.
Revealing mistake: At about the 6:40 mark into the episode, The Bookworm puts an explosive device in the Batmobile which is detected automatically. Batman pushes the auto eject to throw it out of the car. As the special effects prop explodes, you can see the rope it is hanging from as it swings to the side.
Factual error: During the fight to regain his box, the Joker picks up an orange colored deep well drill bit to hit Batman. First, such a device has sufficient weight that one regular man couldn't lift it easily, let alone swing it. Second, the drill bit wouldn't be lying around in the water works guard office.