After Batman takes down Zsasz in the mansion, Dr. Young kneels down and begins hitting him with her left hand. In the cut scene immediately following, Dr. Young is now hitting Zsasz with her right hand.
Batman puts explosive gel on his knuckles and punches Joker - at the very least, his arm should be broken. Though the final cut-scene implies it did.
Harley Quinn could easily fit through the bars of the cell Batman leaves her in, but only in the cut scenes. When the actual game resumes, the bars are a normal length apart.
At the beginning Harley Quinn set charges to an elevator and causes it to crash down while being pulled up by the counter weight.
If the elevator cables were cut (as they clearly were) the elevator comes down, but so should also the counter weight.
The Cryptographic Sequencer is supposed to bypass security protocols. Instead, it destroys whole devices.
The cages in the Aviary lower on their own.
In the opening cutscene, a sign to Arkham can be seen while Batman is driving the Batmobile. The sign is on the left side of the road. In the United States, all signs are on the right side of the rode.
In the opening cutscene, a sign to Arkham can be seen while Batman is driving the Batmobile. The sign is on the left side of the road. Arkham Asylum is located off the coast of Gotham City. According to William Safire, "Gotham City, as all Batman fans know, is New York -- particularly New York below 14th Street, from SoHo to Greenwich Village, the Bowery, Little Italy, Chinatown and the sinister areas around the base of the Manhattan and Brooklyn Bridges." This means that Arkham Asylum is in the United States. In the United States, all signs are on the right side of the rode.
The Batman can somehow rip grates out of the wall. The bolts that are used probably can hold hundreds of pounds, so it is more likely that he would have some sort of tool to deal with them.
The setting is in the United States, yet the measurements are all in metric.
The file storage room above the entrance of Quincy Sharp's office has no doors. It can only be reached by someone with a grappling hook (or a ladder, for that matter).
After defeating Bane, Batman activates the Batmobile, still parked in Arkham North. When Bane surfaces and tries to kill Batman, we see the Batmobile barreling down a hill towards him. This means the car would somehow have to go through two locked doors, which appear in one piece and still locked later in the game, in order to make the journey to Arkham West near the water where Batman and Bane were.
The Riddle flat out tell Batman he has somehow "hacked into" Batman's coms, and yet not only does Batman never caution Oracle about using his real name (over those same "hacked" coms!) in their conversations, they both frequently address each other by their real names as well as their aliases throughout the course of the story, thus fully revealing their identities to The Riddler. Batman would never be that careless.