When Kate Harper and President Josiah 'Jed' Bartlet are discussing the train derailment between Haverhill (pronounced "Haive-rill"), Mass. and Exeter, N.H., Kate pronounces "Haverhill" like halve-er-hill, which Bartlet, being from New Hampshire, would have at least grimaced at, if not corrected her outright.
Railing against how cold it will be outside on the dais during the inauguration, Abbey Bartlett asks "Who in his right mind decided that January would be the right time to hold an outdoor ceremony north of the equator?" President Bartlett replies "Jefferson, Adams, Franklin". President Bartlett, being a fan of historical minutiae, would know better. Inauguration Day, was on March 4 from 1793 until 1933. It was changed to January 20th by the 20th amendment.
When ex-President Bartlet is returning to Manchester, NH in Air Force One (from Washington DC), the plane is shown over an open expanse of water. Since Air Force One has priority clearance, the direct flight from Washington to Manchester would not be over any open water, such as the Atlantic Ocean or even Chesapeake Bay.
On January 20, the sun sets at about 5:15 pm. It would not be possible for sunlight to be streaming in the windows of the Oval Office or of the Bartlet Airplane at about 6:00 pm on the day of the Inauguration.
When Abbey asks Jed who came up with the idea of having the Inauguration on January 20, Bartlet replies "Adams, Jefferson, Franklin". It has already been mentioned that Inauguration Day was March 4 until the 20th Amendment. In addition, neither Adams nor Jefferson were at the Constitutional Convention (Adams was the ambassador to the British, and Jefferson to the French).
After CJ exits the gate, the north lawn of the White House is shown without any setup for the Inaugural Parade. The President's private booth, or any of the bleachers lining Pennsylvania Ave., would've still been in place.