Plot hole: Lee and Dawn overstay their tourist visa in the US, then fly back to the UK for the Office Christmas party. They then get into a taxi to go out to the airport in order to return to the US. They're kidding themselves. As I know (to my cost) when you overstay a tourist visa, the US Immigration Service finds out when you check in at the airport on departure. They'll be refused reentry, and probably barred from entering the US for up to six years. (I got a three year ban.) Neither Lee nor Dawn could possibly be unaware of this nor think they could beat the system as they would have been arrested when they checked in in Florida on departure (as I was), and they would have remained in small holding cells until the paperwork banning their return had been processed and their passports stamped 'Invalid for U.S. Entry', as mine was. (All this takes about seventy two hours, so they'd have missed the Christmas party anyway.) The mistake is NOT that they were able to leave the USA - that is explained (it's still not possible, but in the context of the show and the suspension of disbelief we are meant to believe it) the mistake is that they are on the way back to the airport to return. That's absurd. They would have been under no illusions as to their banned status re reentry to the USA.
Continuity mistake: In the scene with David and Gareth in David's office, David gives a hole-punch to Gareth as a souvenir. The hole-punch has a guide accessory sticking out of it (to punch holes in the middle of different sizes of paper), which is adjustable. At the beginning of the scene it only sticks out by about 2 cm, but by the time David hands it to Gareth it is about 5 cm - even though no-one has touched it. (00:15:35)
Continuity mistake: Often during the show, the viewpoint will switch between cameras to show people from different angles. The problem is, the show is meant to be seen as a documentary, and filmed "for real", meaning these camera switches would reveal the camera that must have been taking the previous shot. No cameras are ever seen, and as a "documentary" it can't be said that the filmmakers could ask the characters to stop in mid-conversation while they moved the previous camera out of the way.
Revealing mistake: Many shots show the trees and bushes in full bloom despite the fact it's Christmas. Most notable is the tree outside the pub window in Brent's second date and the bushes outside Brent's window when he's driving around in the daytime. Acknowledged in the DVD commentary.