Factual error: When Leo is reminding the Congressional Aides about their bosses' drug scandals, he states that Vicodin is a Schedule II Controlled Substance. It is actually a Schedule III Controlled Substance. Given his experience in this area he'd know better.
Character mistake: Both in the title and throughout the episode, people refer to 'Lord John Marbury'. However, the first names of peers are not used when also using their titles; so Marbury should be called either Lord Marbury (more accurately Earl Marbury), or just John Marbury. A minor error in the real world, but the White House is supposed to be aware of diplomatic eccentricities like this.
Character mistake: CJ says that Marbury was ambassador to either India or Pakistan while briefing someone on the phone. Britain doesn't have ambassadors to those countries - being part of the Commonwealth they have High Commissioners instead. A minor error, but one the White House - versed in diplomatic protocol - would be aware of.
Other mistake: I call this one an "Escher Mistake". As Donna and Josh near the end of their chat about the chair that needs repairs, they pass an office door on their left (Nancy McNally's office) just before going through another doorway. The camera that follows them passes "through" a solid wall and emerges in an office, facing a hall and stairwell as Josh and Donna enter from our left as if this has all been a continuous shot. But the camera-through-the-wall moment is actually a cut so that the actors can be on a totally different part of the set. This would be fine, except the staircase they are about to turn left to and climb only rises up about 3 feet, then the user turns left again and heads to our left - but that sends the user into McNally's office, and does so 3 feet off the floor, yet they are seen entering the press room instead.
Character mistake: Sam makes a joke about Laurie being "free of cataracts" because she smokes so much pot. She laughs and says "I get that. That's funny." Except there's nothing to 'get', because Sam is wrong. Cannabis has been used as pain relief for people suffering from glaucoma; it does nothing whatsoever for cataracts. It's surprising that neither Sam nor Laurie would have known this.
Continuity mistake: When Toby is walking back to his office from the press room with his ex-wife, the ex-wife's tag and chain keeps changing through the entire sequence. Sometimes it's over her jacket and the tag's just hanging or it's under the jacket and the tag is sitting just inside her jacket.
Character mistake: In this episode, the character Jeff Breckenridge meets with Josh to plan his Senate confirmation. When explaining what led him to be a civil rights lawyer, he tells how his great grandfather was captured into slavery, and "taken to New Guinea to be shipped to America". The character probably means Guinea, or Guinea Bisseau, or Equatorial Guinea, three places in the Western coast of Africa from which black slaves were sent to America. New Guinea, on the other hand, is half-the-world away, near Australia.
Continuity mistake: After President Bartlett sits down (while speaking to CJ and the staff) he removes his glasses. The shot then changes to one from his back where he's wearing glasses. When the angle changes to the original, he's not wearing his glasses again.
Factual error: In the Don't Ask Don't Tell meeting, the USAF Major is wearing a JCS service badge on his right. That is the position for females. It would actually be worn below his ribbons. He is also not wearing a name tag on his blue shirt after he takes his jacket off. That is the one mandatory item required on his shirt.
Factual error: During the face-off about religion/politics, they get into a shouting match about commandments, both groups being equally convinced that "honour your father and mother" is the 1st or 3rd commandment, respectively. In actual fact it's the 4th or 5th, depending on which version (Catholic or Protestant) of the 10 commandments you go by. See http://www.positiveatheism.org/crt/whichcom.htm.