When Batman and Robin drive the Batmobile onto the dock to reach the Batboat, they leave their car facing out to sea. When they return to the vehicle (after the buoy attack) the Batmobile is facing back towards land.
At the press conference following Batman's fight with the shark, Batman replies to a reporter's question about the yacht by saying, "Nonsense - how can a yacht simply disappear?" Discussing the yacht in the next scene at the Batcave, he says to Robin, "As you, yourself, stated, a yacht simply can't disappear." Batman credits Robin with saying something at the conference he actually said.
When Bruce Wayne and Miss Kitka are kidnapped from her borrowed penthouse apartment, she is wearing a pink lounging gown. Later, at the hideout, when Bruce is allowed to see her, (presumably a prisoner like himself, without access to clothing changes) she is wearing a white one.
The dehydrated members of the security council were of different sizes of sand when they were dehydrated. However, when Batman & Robin rehydrate them - all the sand is of equal volume. It changes back to unequal when they are in their chairs at the UN building.
While Batman is wrestling the Shark on the Bat Ladder, he attaches his bat radio to his utility belt. As he is kicking the Shark, the Radio falls off, presumably into the ocean, but we see it used in many shots thereafter.
While traveling with the Penguin in the Batmobile, the close up shots show that The Penguin is clearly seated behind Robin. On the long shot, when the car "stalls", The Penguin is sitting in between Batman and Robin as there is NO back seat in the Batmobile.
While escaping from the criminals lair, Bruce dives into the ocean and begins to swim away. Upon subsequently returning to Wayne Manor, he and his clothes are completely dry and show no signs of having been wet.
While traveling with the Penguin in the Batmobile, the background road is wide, with highway stripes, and not along the coastline. But exterior shots show the car on a narrow, no-striped road along the coastline.
As the Batcopter approaches a yacht, Robin takes a device to get a bearing on the vessel, in the interior copter scenes. But in exterior shots during the same sequence, Robin is obviously not holding the device.
Just after the Penguin's submarine fires the third torpedo at Batman and Robin on the buoy we hear it explode. Then the sub surfaces to view the destruction. However, in the next shot only the periscope is visible above water.
When the Bat-Copter is on the ground, there are two struts leading from the body to the helicopter to the middle of each of the wings. When in flight, there is an additional cable from the body of the helicopter to the wingtips.
When Batman and Robin take off in the Bat boat for the first time, the spinning rotors from the camera helicopter become visible and stay there for a while until the helicopter catches up with them and the camera pans back down.
Early in the film, Batman establishes that the US Navy sold a surplus pre-atomic submarine to the Penguin. Later, we see this submarine firing a Polaris missile. Pre-atomic (e.g., diesel) submarines were not equipped to fire Polaris missiles.
When Batman and Robin are climbing up the wall, you can see the strings used to hold their capes up, as the scene is clearly a rotated shot of the duo walking horizontally. Although this has been widely regarded as a mistake, the special edition DVD commentary with Adam West and Burt Ward explains that it was done purposely to have some fun for the adult audience.
Batman and Robin solve a pair of the Riddler's riddles that indicate that Catwoman is part of the conspiracy, and the caped crusaders, the Commissioner, and Chief O'Hara seem shocked. Yet earlier in the movie, they had already discussed the possibility that four super-criminals, including the Catwoman, had joined forces.
In the scene where Batman and Robin are trapped on the buoy with a strong magnet pinning their utility-belts to the buoy: After the third torpedo explodes (after hitting a dolphin) we next see Batman and Robin back in their Batboat and escaping, but how did they free themselves? There is no indication that the magnet had shut down (the buoy was never hit, nor did we see the penguin turn it off, or Batman using some Batdevice to shut it off).
From the dialogue, it is obvious that Batman and Robin have met and captured each of the villains before--specifically, they have met, captured, and unmasked Catwoman before. Yet neither of them recognize her as Miss Kitka, despite her not using anything vaguely resembling a disguise.
When Catwoman threatens to have Kitka killed, she says something along the lines of: "Don't come at us, or your sweet Ms Kitka dies." She says that to Batman. From the Catwoman perspective, Batman never met Kitka. Bruce Wayne did.
During the BatCopter flight, Batman instructs Robin to drop BatLadder, which is jettisoned automatically by pressing a button. After Robin hits the button, he states BatLadder away. When the camera cuts to the exterior shot of the BatCopter, we can see Batman unhooking the ladder by hand and then dropping it.
When Penguin gasses the security guards with his umbrella, the guards fall towards each other and stop. The guard on the left's right arm defies gravity to stay at his side rather than, if he were truly gassed, hang loosely as does the guard on the right's. When Penguin returns back past the two guards, both of their arms are hanging loosely.
When viewing the super-criminals currently at large, the pictures shown are still shots taken from the Batman movie trailer. This is especially noticeable for Catwoman, whose picture is taken in the living room of Wayne Manor.
During the Batman/Robin boat scene where they are attempting to find finger prints, Robin claims they are the only ones on the ocean. However moments before if you look in the background some sort of vessel moves by.
As Batman is being lowered into the ocean a shark attacks his leg. When Robin in the Batcopter pulls both Batman and the shark up out of the water, the shark has gallons of water pouring out the back of it where there must have been a large hole.
When Robin turns on the re-hydrating machine, the water in it is blue, however the water running through the piping into the test tubes is clear. And none of the water is emptying into any of the test tubes.
During the fight scene on the submarine, a large wrench is used first by one of the henchmen, then by Batman. It is obviously made of rubber or plastic as it bends as it swishes through the air. Also when it lands on the deck of the submarine it is slightly curled. If it were metal it would not have flexed in this manner.
When Catwoman throws her cat Hecate at her bickering colleagues, the cat lands on the table snarling, with its back arched and darting its head. In the next cut, as the three men cower before the cat, it is clearly sitting and relaxed.