(at around 1h 22 mins) When Batman shows up at Chase's apartment, he comes in through her balcony. It is pouring rain outside, but Batman is totally dry.
(at around 1h 40 mins) When Batman flies out in the Batwing, at first there is no rear wing. In the next shot, the rear wing is there.
(at around 30 mins) 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.
(at around 1h 4 mins) When Bruce is speaking into his video watch (when Alfred tells him the Batmobile is missing), each time the watch is seen in closeup, Bruce's finger moves up the side. It must have been filmed only once, then reused, with the video of Alfred added later.
(at around 53 mins) 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.
Boiling acid does nothing to change the chemical's pH measure. As with any other liquid, boiling any acid causes it to turn into a gas. This can harm people as it creates toxic fumes that can cause severe chemical burns, but the audience never sees the security guard or Batman affected in that way.
(at around 9 mins) There is no way that Batman's grappling hook could not only pierce through solid concrete but also support the weight of a two ton safe.
When Robin is threatening to leave Wayne manor, Bruce looks over his shoulder and says that he's nearly out of gas. Old motorcycles didn't have fuel gauges: if you ran out of fuel you just switched to reserves which would give you another 20 miles or so.
When the news is on television about how Harvey Dent burns the side of his face in a court trial, the amount of acid that is thrown at him does not match the wide and cartoonist scars he later has on the entire left side of his face and neck.
(at around 16 mins) 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.
(at around 1h 30 mins) 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.
(at around 56 mins) 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.
No one ever questions how Bruce Wayne was able to overcome Two-Face's thugs at the circus. However, in some continuities, including Batman: The Animated Series (1992), Bruce Wayne is shown to be a competent fighter/martial artist who attends gyms/dojos/etc. It's likely that's the same case with this movie, and if so, his strength/fighting ability would be public knowledge.
Joel Schumacher's Batman movies are not set in quite the same universe as Tim Burton's, so differences between them (in the back stories of Bruce Wayne and Harvey Dent, for example) are not goofs. (Despite having actors Pat Hingle and Michael Gough playing the same roles).
(at around 52 mins) When The Riddler and Two-Face meet for the first time Riddler can be seen Jim Carrey in his green costume in the background, waiting for his appearance. Just as Sugar says: "I made your favorite tonight" look to the right side of the screen. Not a goof as he was eavesdropping on them talking as he already had his brain boxes in Two Face's lair, which he didn't have on him when he made his entrance.
(at around 1h 30 mins) A pole can be seen on the giant bat as it comes toward young Bruce.
(at around 57 mins) When The Riddler and Two-Face are committing their first robbery (at the jewelry store) the Riddler breaks a 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.
(at around 10 mins) The eye-drop ad explodes before Two-Face's Yin-Yang helicopter even crashes through it.
(at around 1h 24 mins) 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.
Obvious stunt double for Batman when he fights the goons at the gala.
(at around 42 mins) Reflection of the lighting equipment on Dick's motorcycle helmet when he attempts to leave Wayne Manor.
(at around 1h 1 min) The Bat-computer reacts to an intruder in the Batcave by turning the power on for everything, and unlocking the various Bat-vehicles. This response actually makes it easier for an intruder to steal the Batmobile, than if the system did nothing.
(at around 17 mins) When meeting Batman at the Bat-signal, Chase Meridian says "Last night at the bank, I noticed something about Two-Face: his coin." Two-Face never uses his coin within view of her at the bank, and she could not have seen it from the ground if he had. This scene was originally written to take place after the circus scene, where she really did see him use his coin. The bat-signal, seen from Wayne Enterprises, was actually supposed to be what led Bruce to the bank in the first place.
For some reason Chase never questions why Bruce just disappeared during Nygma's party after Two-Face shows up.
(at around 1h 3 mins) A clipboard can be seen on Dr. Meridian's desk with the word 'schizophrenic' misspelled as 'schizofrenic.'
Towards the end, Batman's riddle suggests the myth that bats are blind, which isn't literally true.
When the news is on television about how Harvey Dent burns the side of his face in a court trial, it doesn't make sense for Batman to be there in bright daylight (he only comes out at night), in addition to being allowed to be present to a masked vigilante at a court trial.
Two Face's motivation for wanting to kill Batman makes no sense, Batman didn't know they were going to throw acid in Dent's face, court security should check people before they enter the room, plus Batman shouldn't even being present in a place like a court trial being considered a masked vigilante and in broad daylight (he only goes out at night).