When The Riddler breaks into Two-Face's hideout, he only has his cane with him, but a few minutes later, when he demonstrates how "the box" works, the two green receivers that it uses are already there.
When Bruce is speaking into his video watch (when Alfred tells him the Batmobile is missing), each time the watch is seen is closeup, Bruce's finger moves up the side. It must have been filmed only once, then reused, with the video of Alfred added later.
Before Bruce arrives at Chase's office to show her the riddles he had been receiving, we see him get out of his car, walk up the stairs, and break Chase's door down. He does not have the riddles in his hand until he is ready to ask her about them, and they are way too big and bulky to have fit into his pocket.
When The Riddler tells Two-Face the bargain, Two-Face gives him the "heads, we accept", and "tails, we decline" speech. He then tosses his coin into the air. We then get a close-up shot of the coin. But when it is coming back down, it is clear that it lands on tails, not heads. But, traditionally a coin flip involves catching, covering and flipping the coin again, so we see heads down, which would flip to heads up and Two-Face accepting.
When Bruce uses the torpedo to access the batcave, he travels at nearly 200 mph for almost a minute, indicating that the batcave is some distance below Wayne Manor. But when Dick finds the batcave, it appears that it is only a few stories below. Bruce is traveling from his office in downtown Gotham, which is quite some distance form Wayne Manor, which explains the speeds at which he travels to get there.
Young Bruce Wayne drops his father's diary when he falls into the cave, but lands with it in hand. This may be a deliberate, as the scene is a subjective view of a confusing, traumatic moment, when memory can become unreliable.
When The Riddler and two-Face are committing they're first robbery (at the jewelry store) the Riddler breaks an glass display case and grabs a very big diamond and says: "here's a good one". However when he holds it up to look at it, the frame changes and it has suddenly been reduced to a tiny diamond barely visible. Then Two-Face shows him a big diamond (probably the one we saw earlier) and says: No, No, there is a good one". Jim Carrey apparently picked up the wrong diamond, and in the next frame given it to Tommy Lee Jones. It is also obvious that they tried to hide the frame switch in the editing of the film.
After Batman saves the guests at the Nygmatech party, a newspaper is shown with headlines of the incident, but the actual text of the stories is about the trial of the murderer of Bruce Wayne's parents.
The goof items below may give away important plot points.
After Bruce Wayne gets shot in the head by Two-face and then wakes up in bed, there is a monogram (B.W.) on the pillow to the right of his head. When we cut back to him, after Alfred speaks, the monogram is missing.
During the climax, Dr. Meridian is bound with rope around her midsection, pinning her arms to her side. However, in the shot where she is first dropped, the rope disappears. It then reappears after Batman dives to rescue her.
During the zoom-in on Robin's face as Two-Face falls at the film's climax, it is possible to see two small pieces of damage to Robin's mask, most prominently a vertical nick in the rubber below his right eye. After Two-Face's hand has disappeared below the water, the scene cuts back to Robin and the damage has gone.
Chase Meridian loses her shoes when she gets dropped out of the Riddler's holding tube; her hosed feet are plainly visible in several shots. When Batman and Robin finally pull Chase to safety, she still has no shoes on. However, a few shots after that, she can be seen wearing a pair of black sneakers.
The total fall sequence, from the time the Riddler pushes the button, to the time Chase and Robin are caught by Batman, is forty seconds. In order for this to happen, Chase and Robin would have to have fallen from a height of over six thousand feet, and not the two hundred feet or so we could see from the Riddler's hideout to the rocks below.