Factual error: Sam says the "state-of-the-art" oil tanker he recommends can carry 2.2 million gallons of oil and weighs 308,000 tons. The Exxon Valdez could carry over 60 million gallons (1.48 million barrels) and weighed only 211,500 tons. The writers mistakenly swapped 'gallons' for 'barrels' so that this enormous tanker carries very little oil. It would actually carry 2.2 million "barrels", which equals 92 million gallons.
Factual error: In the scene where the Secret Service chief is in the car with Bartlet and he suddenly realizes Bartlet's injured, he yells to the driver to get to GW Hospital and the limo does a screeching U-turn on what appears to be the Arlington Memorial Bridge. At the time they were supposed to be heading from Rosslyn, VA, where the event was, back to the White House. But GW is in DC, not that far from the White House, so although the U-turn looks cool, it doesn't make geographical sense. Also, in that one shot of the outside of the car, it appears to be pouring, while back at the scene of the shooting as well as ahead of them at the hospital entrance, it's not.
Factual error: In the situation room, Colonel Chase informs Leo that a CH-47 Seahawk helicopter was dispatched from the USS Monterey. Problem is, the CH-47 is called "Chinook" while the SH-60 is the Seahawk (of which the USS Monterey carries two). See: http://picasaweb.google.com/agbeko.Dzamesi/Aircraft/photo5029495094968996146 for photo of an SH-60B Seahawk landing on the USS Monterey.
Factual error: In the scene when Leo tells president Bartlett that he is attending the concert, they refer to the Icelandic ambassador as a man. Leo says his name is Vigdis Olafsdottir, when is fact it is a woman's name that means Vigdis daughter of Olaf. During the concert the ambassador is portrayed by a man. They "borrowed" this name from Tom Clancy's novel Red Storm Rising, by the way.
Factual error: Leo asks how long it will be before the missile system being tested will hit the target, and is told "two minutes, ten seconds". He decides the President should see it, and leaves to fetch him. But doing so results in them both arriving back in the room exactly 3 minutes later. 7 seconds after that, they are told there are still 50 seconds to impact (but it should have occurred 57 seconds ago). After the 50 seconds pass, Leo states the impact will occur in 20 more seconds, and the moment of expected impact does pass at that time. In all, 4 minutes, 16 seconds pass from when "2 minutes, 10 seconds" to impact was announced.