Continuity mistake: In the episode 'Yesterday's Enterprise' (season three, episode 15), watch the final scene with Geordi and Guinan in Ten-Forward. Although time and the universe has returned to normal, Geordi is still wearing the uniform from the alternate timeline (note the sleeve detail, and the extended neck piece).
Revealing mistake: In the scene where Barcley reappears on the transporter with the crewmember, they both fall to the ground. The transporter chief had put up a forcefield around the transporter chamber, in case something goes wrong. However, when the crewmember falls to the ground, his foot goes right through where the forcefield is meant to be. It is still active, as only a few seconds later we see it being deactivated.
Visible crew/equipment: Near the start, just as Riker says "You have all the time you need.", the engine consultant stops and turns around. As he does so, look above his head and you can see a microphone hanging down. The camera quickly adjusts so it's not visible any more. [Not visible in the remastered Blu-ray version].
Other mistake: About two minutes into the episode, in a scene where Riker and Picard go to meet admiral Jameson in the transporter room, just as they are about to enter the turbolift Picard opens his mouth really wide as if making a funny face to someone inside the turbolift. Although it was filmed from the side it is quite easy to spot.