When Toby and Congressman Willis are talking after the meeting, there is a soda cup in front of the congressman. The camera cuts away and when it comes back, the cup has disappeared.
When Josh is listening to "Ave Maria" he claims that Schubert was crazy. Medically speaking, this was untrue and it's more than likely that the writers were thinking of composer Robert Schumann, who did in fact end up in an insane asylum towards the end of his life.
Bartlet mentions that his daughter will be starting at Georgetown University soon and "is scouting for off-campus housing". Georgetown requires all first- and second-year undergraduates to live on-campus, providing an exception only if the student is married, over the age of 21, has dependents, or is living with his/her parents.
Prop mistake: As Bartlett talks about Zoe before the chili dinner, he gives a speech. The ice cubes, plainly visible, are resting on the bottom of the glass. This is proof that prop (fake) ice cubes were used for Bartlett's glass.
The president says there was a 200 billion dollar debt increase despite a budget surplus of 22 billion, a difference of 188 billion. The math indicates it would be a 178 billion dollar difference. Irrelevant as he was lying about all 3 numbers.