When Batman and Robin pull grappling hooks from their utility belts, it zooms in and they are made of wobbly rubber. Whey they throw the hooks in the next frame, it makes a "clink" noise of steel hitting steel as they grab.
When in the space rocket, Mr. Freeze says "At 30,000 feet, your heart will freeze and beat no more!" Obviously, this is false as the human heart can survive that altitude without freezing and the first thing to freeze would be his limbs, not his heart.
When the Dynamic Trio is trying to unfreeze Gotham and can't wait the five hours until dawn, Robin suggests they use the satellites to reflect the sunlight "from the other side of the equator". This in and of itself wouldn't accomplish anything; the equator is a measure of latitude, while time of day (or night) is determined by longitude.
When we first see Dr. Pamela Isley , she says, "Drats! My experiments to mate the Dendrobium orchid and the South American rattlesnake have failed again!" The orchid shown is a Paphiopedilum (slipper) orchid, not a Dendrobium.
When Batman, Batgirl, and Robin leave Poison Ivy's Turkish Bath hideout and go after Mr. Freeze, their outfits all change. Batman gets a new bodysuit with a bat logo that takes up the entire chest, Robin gets a suit with a white bird insignia (as opposed to the red on his other one), and Batgirl acquires a Batman-style mask. With time being of the essence it's not likely they would have gone back to the Batcave to change outfits.
Before and during the final confrontation at Gotham Observatory, Robin is wearing a batsuit with a silver "Robin" logo across it. But when they return to Wayne Mansion and Batman inserts the blue phials into Alfred's drip, Robin's wearing his old batsuit with the red 'Robin' logo on it.
In the final battle when Bane attacks Robin and Batgirl, he grabs both by the throat and pulls them backwards away from the ledge, in the very next shot however his is pushing them forwards into a wall.
When Poison Ivy and Bane arrive in the limousine, Ivy removes the top of the box that she had placed on her lap, and her brown gloves are clearly visible. In the next shot, her gloves have disappeared.
When Barbara is looking at the CD-ROM that Alfred gave her, she looks at a photo of her mother. It has nothing written on it. She then flashes back to her memory of a different copy (Alfred's), which has "Love, Peg" written on it.
It is revealed that Batman has a credit card, which would mean that he would have had to reveal his true identity as Bruce Wayne. Which means someone knows who Batman really is, defeating the purpose of his Batman persona.
During the standoff between Ivy and Batgirl, Robin emerges briefly from the pool of water, before being pulled back in. This footage was obviously not filmed at the time but added later on - its actually footage of Robin getting out of the pool once the fight is over which has been played then reversed to make it seem like a full movement.
The goof item below may give away important plot points.
Near the end of the film Batman stands over Mr. Freeze as Freeze accuses him of killing his wife. Batman tells him that he didn't kill her and pulls out a small video monitor and shows Freeze the footage of Poison Ivy admitting that she was the one that pulled the plug of Freeze's wife. The problem is that Poison Ivy revealed this to Batgirl during their fight and Batman was no where near Poison Ivy at the time to record this admission. In fact, Batman was on the other side of the room, upside down and struggling with the vines that had wrapped themselves around him. He didn't even have a camera in his hands at the time, because he was too busy using the Batsaw to cut the vines away. And even if he had been recording Poison Ivy there's no way the footage would have as smooth as it was on the video monitor the way Batman was struggling to free himself.