When the student comes to Walter's office to hand in his assignment, the first shot of the student standing in the doorway next to the bookshelves shows a book called "Numerical Recipes," which is a well-known text in computer science and, thus, rather out-of-place in Walter's office. Next to it are both editions of a distinctly esoteric book called "Applied Cryptography" which definitely do not belong. In all subsequent shots these books have been removed, suggesting a re-shoot of the first scene.
In New York airport, the flight was announced as "flight to Syria", although it should actually name the airport (Damascus), not the country. But more importantly, there is no direct non-stop flight from New York to Syria.
When Walter gets his first djembe lesson, Tarek explains that all Western music is in 4/4 time (which is not true). He then says djembe music is in 3/4 time (also not true), and he proceeds to teach Walter his first rhythm, in syncopated 4/4 time.
After the announcement at the airport about their flight about to board, Walter and Mouna go to the security control area. Normally passengers would have already gone through security at that point, rather than casually walking towards there after the announcement for the flight departure is made.
As Mouna (Hiam Abbass) and Walter (Richard Jenkins) are becoming closer, Mouna calls Walter Richard - his real name - rather than the character's name.
There is a drop-bolt lock on the inside of the entrance door to Walter's apartment building, but there is no corresponding latch on the door frame.
During the night shot of Manhattan from New York Harbor, the wake of the camera boat is visible in the lower left hand corner.
The deportation center is supposed to be in Queens, but the street signs on the corner are obviously those of Manhattan - not only does it refer to W 115th St (there is no "W" or "E" in Queens), but it's also the telltale green/white color scheme of Manhattan street signage.
The color scheme for the street signs for all areas of NYC was standardized in the mid '90's. All street signs in NYC are now green/white, regardless of location.
While there is no W 115 St in Queens, there are streets in Queens prefaced by E and W (East and West). The southern end of Cross Bay Blvd, in Zip Code 11693, there are about 20 streets that are prefaced E or W.