Jack's truck has an overhead pull line for his air horn (seen on-screen when the Puma truck cuts them off). Early in the film, when Jack is pulling into the Chinese market, you can clearly see that he's *not* pulling that line when the truck's horn sounds.
When Jack, Wang et al exit the Wing Kong Import/Export Exchange Company, Uncle Choo is waiting outside in a yellow bus. Wing Kong security then open fire on the bus as it drives away. However, as the bus turns a corner seconds later you can see that there aren't any bullet holes or broken windows on the side of the bus they were shooting.
When Jack Burton is explaining what happened in the alley, he mentions that Lo Pan had "light coming out of his mouth". However, Jack could not have seen this, as he was already temporarily blinded by the light from Lo Pan's eyes, and had his hands over his eyes by the time the light came out of Lo Pan's mouth.
Jack and Wang Chi change places as they chase the punks at the airport parking lot. When we see them from the back Wang is on the right (close up on back of his jacket. Then we cut to a frontal view and he is still on the right of the screen, while he should be on the left.
During the first stage of the fight between the Chang Sings and Wing Kongs in the alley, there is a Wing Kong firing a Thompson Sub-Machine Gun. While he's firing, he's clearly using the model with the double handles. However when he gets shot and drops the gun, the model changes to the version with the flat bar grip in the front.
When Kurt Russell is pulling himself up on the wheelchair from falling into the well, you can clearly see an effects hinge under the chair that is actually pulling the chair back up, even though it appears as though Jack Burton is doing this with pure strength.
When Egg serves his magic potion to Jack and the rest of his warriors in Lopan's bar, you can see that his goblet has ice all over its bottom half, showing that they used dry ice to make the goblets steam and bubble.
During her first scene at the airport, the color contacts on Gracie's "green" eyes are sometimes visibly out of place (look at her right eye when she turns to Jack after asking "what are they doing here?").
The goof item below may give away important plot points.
During the wedding ceremony, Lo Pan can clearly be seen grabbing Miao Yin's wrist to prick her with the ceremonial needle. However, the entire purpose of the ceremony is to lift his curse and return Lo Pan from his current ghostly state to flesh-and-blood. He should not be able to grab her wrist until after the ceremony is completed.