Plot hole: Willow and Cordy disappear during a vampire attack. We later see that they spent the entire night hiding out in a closet, but wouldn't Buffy have gone looking for them once the vampires were gone? She knew they had run off somewhere (she mentions it to Giles when she gets a stake from the library). Instead she just goes home, leaving them still missing as far as anyone knows.
Continuity mistake: In the episode (series six) where Buffy thinks she killed Warren's ex-girlfriend, when she's at the police station, in the over the shoulder shot of Desk Sergeant, he is holding the phone between his shoulder and ear. But in the front shot of him the phone has jumped to his hand.
Revealing mistake: Near the end of the episode when the Mayor releases the vampire from his cage and lets him cut his head in half, you can tell that the Mayor's head is a still image when it fixes itself. The expression on his face doesn't change at all and looks completely unrealistic.
Revealing mistake: Right at the beginning of the episode, Buffy is sat outside with Willow and Willow is levitating a pencil. The pencil is obviously CGI as it stands out from the rest of the picture and doesn't cast a shadow. It also disappears from view whenever there is a side-angle shot of Willow.
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.
Continuity mistake: At the end of the episode when Buffy is talking to Giles about how she can't face her friends, her hair is over her ears for the entire conversation, but when she is about to walk away, her hair is suddenly tucked behind her ear and she couldn't have tucked it behind as that side of her head is still in view when the camera's on Giles.
Revealing mistake: The entire scene where Buffy is experiencing accelerated time, shadows of trees move in the wind but never actually grow or shrink. Even over short increments of time, the sun can move shadows noticeably, and judging from the speed of the people around her, Buffy is in accelerated time for at least half an hour.