When the President, his wife and daughter are playing scrabble. The daughter places down Zygote and claims 40 points. A clear shot of the board shows that she should actually claim 60 points Z=10, Y=4, G=2, O=1, T=2 (1+double letter score) and E=1 each: total 20 x triple word = 60
When Anna discusses with her father about being allowed to go to the Love Parade, she says she's not 12 anymore as when he had his first inauguration, but rather 18, so she might decide to vote for him. But he won't be in any more elections, since he's obviously in his second (and thus last) term.
After Anna wipes the s'mores on Ben's face, she slowly tries to wipe it off in a flirty manner. But when the camera turns back at Ben after she wipes it off, his face looks clean, as if nothing ever touched it. Plus, Anna's hand is now clean, too.
In the hotel where Anna's parents are discussing her "freedom", the right side of Mrs. Foster's robe is all the way across her chest. In the next shot it is half way across her chest and the next shot it is not across her chest at all, yet she doesn't move the entire time.
When the news report at the beginning of the film shows Anna being toured around Prague, there is a segment showing Anna having a "photo op" at Prague Castle. In the shot, Prague Castle is clearly visible in the far distance in the background of the shot.
When playing Scrabble, the First Lady players QUIVERS and claims a 50 point bonus. However, that bonus is for using all 7 letters in your rack. But quivers is just a 7 letter word, and she had to play at least off at least one letter already on the board. It is possible that she turned the President's HAT into HATS, but then she should have claimed the score for that word as well. One way or another, there was an error in scoring.
The secret service is "rescuing" Anna from the love parade, which takes place in Berlin. The police car to the left (shot from the helicopter) is clearly visible as a Bavarian police car (painting of the green part of the police car is different in Bavaria to the rest of Germany).
When McGruff gets a tattoo it is ready instantly, anyone who ever had a tattoo can testify that it can take up to two weeks to heal and you DON'T want to walk around in the sunshine with it during that time. But, with the fact that he doesn't have a tattoo later on, it's obvious it's a temporary one, one that can come off easily at the wearer's choosing.
When Anna and Ben are with McGruff, shadows of the electric poles frequently appear behind the bench Anna and Ben are sitting on. But when you look outside through the window, clearly no poles passing by.
At the end of the movie, during the press conference, Anna is seen in the window in the background, and when the camera closes in on her, two panes of glass are clearly missing from the part of the window Anna is standing behind.