In Chapter 1, Batman Takes Over, Batman and Robin are in the Bat Cave still dressed in their caped outfits while examining evidence. Alfred phones to say that Vickie has arrived at the residence to see Bruce Wayne. Batman changes into his business suit while Robin does not change clothing and remains in the cave to examine evidence. Bruce then tricks Vickie by having Robin (still dressed in his cape in the cave) phone upstairs with the message that Vickie must leave to cover an auto accident. When Bruce returns to the cave, Robin is now dressed back in his business suit like Bruce, so they both pull out their caped outfits from a drawer to change together in order to leave and fight crime.
Whenever the Bat Signal is shown, the same cloud formation is behind it.
When the stolen patrol car, driven by the "invisible" Wizard crashes through the police road block, there is clearly someone at the wheel. However, when the car then rolls to a stop, it is seemingly empty.
When the gang lands their plane in Chapter 1, Batman Takes Over, there are windows in the nose of the plane. When the gang takes off in Chapter 2, Tunnel Of Terror, the plane has no windows in the nose.
After Robin is kidnapped, in the truck and all during the outdoor scenes, he never has his shoes on; he is barefoot in only his tights. Later, when in Morton's office, Robin has his boots on again.
In Chapter 10, the office building hide-out has a switch that will render unconscious (i.e., shock) anyone who touches metal in the building, which happens to Batman when he touches a doorknob. This would not work as the metal in a doorknob and its jamb are completely isolated from other metal in the building. The electrical switch that Vicki Vale's brother would have to have been wired to each office door's knobs and hinges for this to work.
Contrary to the Chapter: 9 title "Robin Meets the Wizard", Robin never sees the Wizard sneak up behind him.
In Chapter 8, every box in the warehouse is upside down with "This Side Up" pointing down. However, this is not necessarily a Goof as incompetent gang members likely neglected (or maybe were unable) to read.
Wires can be seen when The Wizard is supposed to be invisible, most notably in the telephone booth when The Wizard is on the run and calls his headquarters.
Batman cuts through a steel door with an oxy-acetylene torch that he took from his belt. There was nowhere on his person where this torch could have been concealed. This was probably scripted to be Batman's 3-inch mini-torch from the comics.
Batman and Robin are in a plane flying at a "high altitude", but when Robin is looking through the binoculars at the bad guys driving to pick up the ransom money, the POV is from ground level.
In Chapter 15, when the policeman is putting the cuffs on Wizard's waiting henchman, the officer can't get the prop cuffs to work. He then holds the cuffs against the wrist of the bad guy and walks him away while holding his wrist.
When the remote-controlled cars are running without drivers, a slumped-over stunt driver can be seen in one of them.
The Bat Signal is shown to be working in broad daylight.
As Bruce Wayne's own private car is used as the "honorary" Batmobile, anyone who wants to trace the Batman's identity needs only to take down his license plate number, as Vicky Vale notices in Chapter 7.
On the phone, in character as Batman, Bruce Wayne begins his report to Commissioner Gordon with "Dick and I...", but to maintain his separation of identities, it should have been "Robin and I".
While Batman is driving to avoid the bales of straw or hay, he avoids the second-to-last one by driving to right side of it. The last bale is on the left side, across the center line; however, he drives to the far left side instead of just going straight.
In Chapter 8, Batman and Robin are tracking the crooks to their hideout. The Geiger counter sound is getting louder. Batman says, "The Geiger counter is getting louder!" Robin then says, "I guess we're getting louder!" He should have said, "I guess we're getting closer!"