While Leo and Admiral Fitzwallace are watching the kidnapper's broadcast of Zoe holding the day's newspaper, the camera sweeps the room to reveal a map of the Middle East. This map omits the existence of the nation of Qumar, showing the real-world political borders of Iran.
In the 5th Season, Ryan Pierce became an intern for Josh Lyman. Ryan claims Franklin Pierce, 14th President of the United States, is his Great-Great Grandfather, however, Pierce never had a son live passed 11 years old.
In the scene where the kidnappers' message is being played in the situation room, the Arabic message is actually Abdul Shareef's dialogue from Season 3 when he met with the president, re-used. The producers simply gave a different translation.
The goof items below may give away important plot points.
In the scene where McGarry and Fitzwallace are told that the FBI think they've found Zoe Bartlet, the clocks on the wall of the White House situation room show a seven hour difference between POTUS time (the time where the president is) and Zulu time (UTC/GMT). However, as the president is in Washington, depending on daylight savings, there can only be a four/five hour difference.
When Zoey is found and as she is still sitting in the back of the ambulance at the place where she was being held hostage, one of the Secret Service agents tells the Bartlets that she has "no signs of sexual assault." The only way to determine this is to do a pelvic exam (i.e., a physical examination of the external and internal genitalia) and to utilize a rape kit, both of which are highly invasive procedures that would never be done at a busy crime scene or in the back of an ambulance; they would only be done at a hospital.