Continuity mistake: At the conference, when the President is announced, there is a wide shot of the crowd and you see a lot of people standing in the background against the wall. There are also several spotlights turned on. Then it cuts to the stage where Frank and the squad are walking. When it cuts back to the crowd all the people in the background are gone and the spotlights are switched off. The crowd reappears later and the spotlights are on again.
Deliberate mistake: When Ed's looking at the drawings done by the sketch artist near the start, he's flipping over a book of them. Trouble is that they all end up facing the camera, meaning that Ed must have been looking at the blank side - we see the blank pages briefly as he flips them.
Revealing mistake: About half an hour into the movie, when Frank is in the tank, he plows into a supposedly concrete zoo wall, which wobbles. Also, the wall splinters where it meets the sidewalk, revealing that the "wall" is actually a wood prop, and a flimsy one at that. Five seconds later, when people and animals are shown running out, the same wall is shown with a different shaped hole and pieces of concrete lying around it. Apparently the concrete pieces aren't too real either, as the camel kicks one around as easy as if it were made of foam.
Revealing mistake: When Frank drops his helmet into the machinery below him, cogwheels start flying. Then it cuts to a closeup of Frank and a cogwheel suddenly strikes the roof right in front of him. However it is obvious that the cogwheel comes through the roof from above and not from the machinery below.
Continuity mistake: When Frank is fighting with Savage in the bathroom, he gets a towel thrown on his head. When he pulls the towel off his hair is messed up. When you see him again a second later, his hair is neatly combed (but with no particular reference to it as a joke, unlike towards the end when he miraculously gets clean and acts surprised).