Multiple references to 9/11 make it clear that this movie is set in the 2000s, but when the world's leaders convene in Kim Jong-il's palace, a sign indicates that the country Zaire is present and its representative looks like Mobutu Sese Seko with his signature leopardskin toque. Mobutu died in 1997 and Zaire was renamed the Democratic Republic of the Congo in the same year.
When Team America is fighting the Film Actor's Guild while trying to get into the peace ceremony, just after Tim Robbins is burned to death, the sound of hands clapping starts before the shot changes back over to the stage.
When the world leaders meet at Kim Jong-il's peace conference the sign above Tony Blair's head reads as England. Tony Blair was Prime Minister of the United Kingdom (which consists of England, Scotland, Wales and Northern Ireland), not just England.
(at around 3 mins) we see the inside of Team Base for the first time. In this scene, and all later scenes in this location, both male and female characters have black eyeliner. Outside of Team Base, the characters do not have black eyeliner.
When Chris fires on the Korean sub, the number on the bow is 1252. The sub that fires on Chris has number 1350 on its bow. However, the next shot shows Chris passing sub #1252 with no damage to its starboard side.
During the scene where the team is fighting the Koreans, after Lisa says, "We lost I.N.T.E.L.L.I.G.E.N.C.E. I repeat, we have no I.N.T.E.L.L.I.G.E.N.C.E", you can see the reflection of a face in the front window of Joe's helicopter during its head on shot.
After his show at the beginning, Gary looks in the mirror as he remembers what happened to his brother. His reflection in the mirror shows him smiling, but the expression on his face shows he is sad and has his mouth open slightly.
When the terrorist pops up from the fountain and shoots Carson in the Paris scene, you can see the Eiffel Tower in the back. If you look closely, it is the bottom half of the tower which was left standing after it is blown up.