When Liu Jian meets Jessica for the first time, he lets her use the bathroom in the shop. When she comes out of the bathroom, she sits down and eats Liu Jian's food with her fingers, yet does not touch the bowl with her mouth. After shes gone, Liu Jian inspects the bowl and see's her lipstick on the edge of the dish yet her mouth never touched the dish.
(at around 9 mins) Insp. Richard's men have just rushed into the hotel room. As the men line up behind Insp. Richard, it's obvious that not only do the "twins" look nothing alike, but the twin on the right is at least six inches taller than the twin on the left.
In the beginning when Liu Jian attempts to escape the French and breaks the door open after going down one level, he closes the glass door into the balcony. But when Richard and his men go into that room the balcony door is open.
When Liu escapes the boat and jump under the bridge, he quickly join a metro platform. The name that can be seen on the wall is "Invalides". When Liu enters the metro train the name can still be seen, but when the metro train leaves the station, the other station name boards on the wall display "Porte des Lilas". (The scene was actually shot in the former "Porte des Lilas" metro station that is now dedicated for the shooting of films. The Director forgot to ask the staff to change all the station name boards to "Invalides".)
The apple on the laptop is upside down because that's how they were designed at the time; the apple would then be right side up when the computer is closed. If you look at the computer you see that it is in fact open and the latch is at the top where it should be.
The goof item below may give away important plot points.
Liu leaves Jessica in the hospital with the acupuncture needle that he used to put her to sleep still in her neck. When he comes back the next morning, she is still asleep and the needle is still in her neck. She is in the hospital because she was shot. Either no nurses came in to check on her during the night, or they were so incompetent that they didn't notice the needle.