Continuity mistake: When Mrs. Peel and Steed are playing chess, some pieces are on the wrong squares.
Continuity mistake: Sir August DeWinter confronts an assembled United Nations Council (or something like it) and makes his demands for money. To prove his power, he unleashes a blizzard on the building. Unfortunately, the Council room has giant windows, and when we look outside, we can clearly see that it is a sunny day with emerald green trees.
Continuity mistake: When Peel and Steed are being chased by the bees, there is a shot looking down the road to their car. As they swerve you can see that there are 3, maybe 4 people in the car when there should only be 2.
Continuity mistake: There is a scene where Mrs. Peel is trapped in the White Room in DeWinter's mansion. Above the fireplace is a mirror. Cut to the outside, where Steed walks. He approaches a window and taps his cane on it, revealing that the windows are fake. They are simply brick or concrete painted to look like windows. Back into the White Room. Mrs. Peel hears the tapping and locates the source at the mirror. She runs and dives into the mirror and it shatters, and she somehow ends up outside next to Steed. Remember that there where no glass windows. Did she break through the bricks?
Continuity mistake: During the chase scene where Steed and Peel are fleeing the robot bees in the blue Jaguar, one of the Jag's doors disappears between shots. We never see the car lose the door or take a hit that might inflict that particular damage.