When Josh is talking to the White House Advanced Scheduling regarding Leo's inability to attend a night meeting, he takes the cell phone from Annabeth and placed the back of the cell phone on his ear. The front face of the cell phone is clearly visible when the camera pans to Josh's left. This is corrected in the next shot.
When the Vinick team is talking strategy on the plane a giant platter of fruit appears on the table out of nowhere between shots.
While in the Roosevelt room at the White House, Bram informs the candidate and the rest of the staff they'll be staying at the Marriott that evening. Later, when Donna attempts to slide her room key to Josh, the mini-envelope that contains the key says "The Capitol Grand Hotel". This is not the Marriott, and is a fictional hotel name.