Continuity mistake: When Riggs and Palmer fall through the roof and onto the bed, the debris under them changes. In the final shot there's a huge piece of the ceiling in between and under them that wasn't in the previous shots.
Continuity mistake: When James is taking pictures from a car as men are entering a hotel, in the shots of him head on he is using a Canon EOS camera with a 70-200mm white lens. In shots from the side he is using a Nikon camera with an older Nikon lens. (00:27:35)
Out of Time - S2-E1
Other mistake: Sara tells Jefferson that the professor has a one hell of a right hook (referring to Martin knocking Einstein out). Jefferson replies, "seriously, Gray punched him out?" The subtitles mistakenly say "Ray."
Plot hole: They only take out the shrapnel from the gut shot - they never deal with the shrapnel in his shoulder at all. (00:04:25 - 00:05:00)
Seeds + Permafrost + Feather - S3-E17
Factual error: Bozer and for that matter I would assume all of the team went to a spycraft academy. A basic subject in any of such academies is surveillance, including proper techniques to follow and stake out someone. Bozer being so inept at tracking MacGyver's father is inconceivable given their training.
Factual error: When the trucker reaches into the bag that's suppose to have his cash with his "bonus", he's bitten by a venomous snake. However, we see the snake and it's a non-venomous king snake.
The King of the Delta Blues - S2-E6
Character mistake: "Are you going to send in the calvary?" is said twice. It should be "cavalry". (00:09:40)
Other mistake: Issac is on the phone with his boss and he is holding his phone upside down, noticeable because his otter box case is upside down. (00:14:41)
Continuity mistake: When Dmitri and the doctor first try to get Vanessa to feed on human blood, the blood bag keeps switching from full to half-empty from one shot to the next. (00:16:05)
Sublimation - S2-E3
Continuity mistake: When Katie is talking to Grace, she has the works of Confucius under the Bible and Koran. When Grace says she likes how people won't die any more, the shot changes and the works of Confucius is now on top.
Episode #1.4 - S1-E4
Other mistake: A major plot point is the rivalry between the Wehrmacht and the SS, who refuse to work together. After the explosion at the cemetery, it is the Wehrmacht who are explicitly described as rounding up and interning suspected resistance fighters, with the SS not being party to it. However, when Archer visits the internment camp, most of the staff are wearing SS uniform.
Continuity mistake: Harry drives away from a crime scene and his rear numberplate is visible as D703 VAN. We then see Harry in his car from the front, and the numberplate has changed to E160 HLT. (00:21:18 - 00:22:31)