When handing Batman the Shark-Repellent Bat Spray, Robin is hanging upside down directly above Batman, but when Batman uses the spray on the shark, Robin has disappeared.
(at around 9 mins) When Robin is handing Batman the shark repellent, the distance between the two of them changes between shots.
(at around 1h 30 mins) When Batman forces the submarine to surface (right before the batfight) the sub is (naturally) wet. However, when the Batboat pulls along side moments later, everything is dry as a bone.
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.
(at around 1h 10 mins) 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.
(at around 29 mins) 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.
In the opening scenes, the running lights on the BatCopter are reversed. Green should be on the right/starboard side and the red should be on the left/port side.
(at around 1h) 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.
(at around 5 mins) While Batman and Robin are flying the Batcopter they fly over women in bikini at top of a skyscraper. They start jumping up and down and one of the woman in bikini has her breast exposed. Its censored.
During the submarine fight at the end, Catwoman pushes Batman into the water. When Batman hits the water, there is another Batman already in the water.
When taking Penguin to the Batcave, there is a point of view shot from the hood of the Batmobile as they drive into the actual Batcave entrance at Bronson Canyon. In reality, that is an entrance to a tunnel, meaning there's an exit. The exit of the tunnel was blocked off, but if you look there's several places where daylight is shining through.
(at around 1h 30 mins) During the final fight scene on the deck of the sub, wrinkles are visible on the sky backdrop.
(at around 1h 15 mins) As Batman and Robin are running across the street, a passenger in a Corvette convertible can be clearly seen looking, smiling and waving directly at the camera crew.
(at around 9 mins) When Batman has the shark on his leg for as long as he does there should be holes torn in he tights. There are none.
Although Batman was correct that somewhere during the many mile ride that the Penguin would use his knock out gas on him and Robin and then steal the Batmobile. He had no ide of the exact location to hide the Bat Cycle which just happened to be hidden at the exact same location.
Someone has already submitted wrinkles in sky backdrop during fight on submarine. There are also waves splashing up against it.
When the obviously fake shark is hanging on batman's leg for so long not only would his tights be torn as someone already submitted but he'd be bleeding with severe damage to his legs. Plus it appeared they also had part of the ladder in the shark's mouth so that would have been chewed through.
(at around 57 mins) When Riddler, Penguin, and Catwoman dehydrate some of their pirate flunkies, The Riddler can sometimes be heard laughing even though his mouth isn't moving.
(at around 1h 2 mins) When Batman is running with the bomb, right after he meets the two nuns for the first time, camera crew are clearly reflected in a window.
(at around 4 mins) When we first see the Batcopter being pulled from the hangar, the whole camera crew and their equipment is reflected in the glass of the copter.
(at around 20 mins) 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.
(at around 22 mins) Wrinkles on the sky backdrop are visible when Robin and Batman arrive at the buoy.
(at around 31 mins) 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 their search for finger prints on the projector buoy, Batman & Robin must have forgotten that all 4 of their arch enemies wear gloves.
(at around 27 mins) When 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).
Why are the Dynamic Duo bothering with trying to find fingerprints on the buoy?? They KNOW who they're dealing with.
(at around 1h 21 mins) 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.
(at around 9 mins) 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.
(at around 28 mins) When Robin contacts the Department of Defense, he say into the phone, "Please give me the Pentagon in Washington D.C." The Pentagon is located in Arlington County, Virginia, across the Potomac River from Washington, D.C.
(at around 1h 24 mins) When the Riddler is hitting all the switches to launch the missile from the submarine, he misses one.