At the end of Chapter 2, as Batman is battling with the thugs, his cape is ripped off and thrown to the floor. After a brief cutaway to Alfred waiting in the car, it is back on his shoulders with no apparent break in the action.
At the end of Chapter 4, Batman blows a hole in the roof of an armored car, but when it goes over the cliff moments later, there is no hole.
When Alfred's fake beard is ripped off by Daka's henchmen in Chapter 6, his chin is beardless, but when he meets up with Batman in Chapter 7, the part of the beard covering his chin has been restored.
In the beginning of Chapter 10, Robin dives off of the ship into the ocean. When next seen, he is dry and caught in the ship's cargo net.
In the fight scene of Chapter 2, the bad guys are all wearing fedoras. Two of their hats fall off but are back on in the next shots of the fight.
In Chapter 5, it's implied that the pilots would have activated the autopilot feature before both leave their seats to fight Batman; however, when the plane is being shot at, it's seen twisting and turning. The autopilot would have kept the plane on a steady course.
The Native American proprietor of the trading post at the gas station hands Bruce Wayne two pieces of black pottery and claims they were carved from rocks taken from the cave. In reality they look to be Santa Clara Pottery that is made from clay that is shaped into the bowls and fired. The distinctive carvings near the rims of the two pieces indicate Santa Clara Pueblo and since these are from the early 1940s, they would be quite valuable today.
In Chapter 5, when the plane is being shot at, the wires supporting the plane are clearly visible.
A hairy-legged stuntman plays Robin (whose legs are clean-shaven) in several shots.
In Chapter 2, Alfred is driving Bruce and Dick to The House of the Open Door. As the car goes through a narrow alley, Alfred is played by a stand-in without a mustache.
Strings are attached to the bats as they fly around in the cave.
At the end of Chapter 7, Batman is thrown down a laundry chute. When Batman hits the ground, a cloth dummy is obviously used.
In Chapter 9, Alfred, in a state of panic, blares out "Master, Master!" to Batman, while Batman, Robin, and others are in the cave. Fortunately for Batman, nobody seems to notice.
In Chapter 1, the narrator explains that Bruce Wayne lives in Gotham City. In Chapter 5, Bruce Wayne's mail has an address in Los Angeles.
As Bruce Wayne's own private car is used as the honorary Batmobile, anyone who wants to trace the Batman's identity only needs to take down his license plate number.
In Chapter 5, the microphone can be seen moving across the top of the screen in the ghost train scene.
In Chapter 8, Robin addresses Batman as "Bruce" when they are both costumed in action, violating the secret identity protocol.
In Chapter 5, Dick Grayson addresses his mentor over the radio as "Bruce". As Batman is in costumed fighting mode, this violates the secret identity protocol.
In Chapter 1, Bruce mentions avoiding his creditors. It's unlikely a billionaire playboy would have creditors. (It may have been intended as a joke, but it certainly wasn't delivered as one.)