Revealing mistake: In "Dead Things," after Buffy and Spike sleep together, when Spike asks, "Are we having a conversation?" and Buffy says, "What? No?" There is a side shot of her. If you look closely at her back (pausing helps), you can just barely see Buffy's skin colored tube top. Trust me, it's there. The time code is 1:22.
Visible crew/equipment: While Buffy, Giles, and the gang are in the library fighting the demon that came out of the Hellmouth, the camera zooms toward the door that leads into the hall, where you can see Xander being chased by the dead guys. As the camera zooms in closer towards the round window in the door, you can see its reflection in the glass when the strobe light flashes.
Deliberate mistake: In the fighting scene, the glove of Myhnegon is used by the new watcher who directs it toward the ceiling of Angel's living quarters where there is a huge skylight. The skylight is not seen in any other episode and its presence would have burned him on other occasions.
Character mistake: Giles picks up the potential slayer Chao-Ahn from Shanghai in Episode 7.12, "Potential". However, when she arrives in Sunnydale (Episode 7.14, "First Date"), the language she speaks is neither Mandarin nor Shanghainese (the two main languages spoken in Shanghai), but Cantonese, which is not widely spoken there.