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.
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).
Continuity mistake: When the guard is reading the tabloid paper, he has it fully open (so that he's holding the back page in his right hand) and starts to turn the page. In the next shot, when he's startled, all of the suddenly the paper is closed so that the "spine" is now in his right hand.