In the first season finale, an F-117 Nighthawk stealth fighter is said to be shot down while patrolling the no-fly zone in Iraq. Although the F-117 is called a fighter, it actually is a bomber with no air-to-air capacity, making it unable to patrol a no-fly zone.
In the opening teaser, President Bartlet says that "a guy on my staff showed me a report...". When the same scene is shown near the end of the episode, he says "a guy on my staff named Charles Young showed me a report..."
Admiral Fitzwallace observes that. The Presidential Seal, has a shield bearing bald eagle clutching olive branches in its right talons and arrows in its left. The eagle's head is turned toward the olive branches. The show mentions that in times of war, the Presidential Seals are replaced with an eagle whose head is turned toward the arrows. This is not true at all. Prior to 1945, the eagle on the Presidential seal pointed to its left, and the arrows. In 1945 it was redesigned to point to its right and the olive branch. The reasoning for the change was that in heraldry, left pointing items were considered dishonorable. In order to "sell" the new seal to the general public, the press release stated that in the time of peace following WWII the eagle would now point to the olive branch of peace.
An F-117 is shot down over Iraq, and it is stated that the plane was conducting routine surveillance when it was targeted and hit. Before it was retired in 2008, the F-117 was a stealth ground attack aircraft; it would never have been used for the kind of monitoring that was described here.
When the F-117 pilot is rescued in Iraq, President Bartlet is told he is safe and on his way to Tel Aviv, Israel. An American military member under those circumstances would have either been taken to a U.S. military location in the region, either a base or a U.S. Navy ship. If he had been evacuated to Israel, he would not have gone to Tel Aviv but to Haifa, which is Israel's largest port and (unlike TA) often hosts U.S. vessels.
At the town hall meeting, President Bartlet claims that his great-grandfather's great-grandfather, Josiah Bartlett, signed the Declaration of Independence. His name is spelled differently from the character on the show, but it's common for names to shift slightly in spelling over so many generations.