Continuity mistake: When Jackie Chan and Chris Tucker first arrive at the museum/fundraiser, Jackie Chan's hair is the same as it was throughout the movie - longish in the back and moussed/gelled back. As the scenes at the museum progress into the showdown with Jun Tao, all of a sudden, Jackie Chan's hair is a lot shorter.
Continuity mistake: During the fight scene at the Foo Chow restaurant, Tucker and Chan join hands while beating up the bad guys. They stop for a moment to look at their handiwork and their hands are still joined, then the camera cuts to a different angle and their hands are nowhere near each other.
Plot hole: The kidnappers initially ask for a ransom of $50 million. $20 million in $20s, $20 million in $50s, and $10 million in $10s. This translates to a grand total of 2.4 million bills (pieces of paper, that is) Every bill of American currency is 2.61 inches wide, 6.14 inches long, and .0043 inches thick. This makes roughly 2,791 bills to form a 1 foot tall stack. This means that the 2.4 million bills would take up a space of 96 cubic feet (assuming they are packed as tightly as possible). Also, since 490 bills make 1 pound, the 2.4 million bills would weigh (interestingly enough) just over 2.4 tons. No way they can move it.
Other mistake: Phillip Baker Hall sends Chris Tucker to meet with the FBI immediately after he tells the agent on the phone that he will be sending someone over to them right away. Chris is told by the FBI guy that Jackie Chan's plane arrives in one hour. After Chris picks up Jackie from the airport, he calls his boss to vent his concern. This is supposedly the same day, but both Phillip and Elizabeth Pena are wearing different outfits.
Factual error: When Carter is meeting with Clive in a diner car park, as Clive is escaping in a car with a trunk full of C4, Carter starts shooting at the car. It eventually blows up leaving Clive badly injured. The explosion is caused by Carter's bullets reacting with the C4. However, C4 is inert and if a bullet hit it, it wouldn't create an explosion. You could also argue that this explosion was caused by Carter shooting Clive's gas tank, but this is another myth created by Hollywood to please audiences.
Continuity mistake: When Thomas (aka Juntao) and Lee are fighting in the ceiling beams near the end of the movie, Thomas's briefcase flies open when he is trying to get Lee to let go. You see a lot of money explode out of the briefcase, and the briefcase falls to the ground. When the shot changes to a semi-bird's-eye view of the floor, the money appears to be all on the floor already. However, when the shot changes back, you see bills flitting about behind Thomas. This happens several times.
Plot hole: After the first attempt by the kidnappers to get the ransom money, they tell the FBI to bring the money behind the Foo Chow restaraunt. Clyde gives Carter and Lee the exact same destination. Why in the world would kidnappers give the FBI a drop-off location exactly where they are based?
Visible crew/equipment: As Chris Tucker rescues the little girl from the van, you can see a bearded guy in a white baseball hat, presumably a crew member, standing a couple feet away, reflecting off the back windows of the van when he opens the door and right after Chris Tucker says the line; "Tell your friends about me." (01:18:10)
Continuity mistake: In the big fight seen in the museum Jackie is holding up a large vase while 2 bad guys are punching him. Jackie sees one of those pole things that hold the velvet ropes and wraps his leg around it. When he does you can clearly see that the the pole is right side up, then when he kicks his leg up to hit the bad guy in the face is it upside down. (01:19:15)