In the episode, "Two Cathedrals", the credit for the character Dolores Landingham is misspelled "Mrs. Landinham".
The President is asked whether he will seek another term in office at the end of the episode by a reporter. The reporter repeats the question.
The question when repeated by the reporter is slightly different when the flash back is shown in the first episode of the next season.
During a flashback where Mrs Landingham is attempting to convince Jed to tackle the pay difference between male and female staff she mentions that "Mr Ryan gets paid an additional $400 a year to coach the rowing team." Jed corrects her use of "team" to "crew." When she asks for clarification of what he means he says "it's not rowing, it's crew." While one might think Jed is correcting her description of a "team," the sport is usually only referred to as "crew," with no mention of "rowing," so he's correcting her description of the sport itself, not the athletes who participate.
During the final press conference, there is a close-up of two photographers. The one furthest away still has the lens cap on.
During Mrs. Landingham's funeral, a camera is visible at around 19:10.
The motorcade passes the National Cathedral on the way from the White House to the State Department. In fact, the State Department is only several blocks from the White House (due west of the Ellipse) while the National Cathedral is more than three miles away. Furthermore, the motorcade appears to pass directly past the front doors of the cathedral at full speed. While there is a paved area in front of the cathedral, the street is almost 200 feet from the entrance and the geography of the cathedral's close wouldn't permit that speed.
When the president is in the oval office, stewing about re-election, the door blows open after he closes all of the other doors and it remains open. There are two problems with this. First, a wind tunnel would not occur when all doors are closed; at least one door into the office would need to be open. Second, there is always a Marine standing at the door to the oval office, yet none are visible and nobody moves to close the door that flew open.