Continuity mistake: In the episode 'Anne' when the latest people to be brought into hell are lined up with a demon talking to them, there are 3 or 4 people standing between Buffy and Lily. When the demon asks them all in turn 'Who are you?' we see him asking Lily and after that the shot rests on Buffy, but in the background we only hear him ask one more person before he gets to Buffy.
Revealing mistake: Both the good and evil Willows' faces are barely seen onscreen at the same time due to them being played by the same actress, but when they occasionally are the good Willow stands out from the picture more than the evil Willow. She must have been filmed separately from the good Willow and edited in later.
Continuity mistake: When Buffy and Spike are fighting on campus during the day, Xander tries to help Buffy but Spike throws him against a lamppost. When he lands in the ground he falls completely on the grass on his side but if you look behind Spike in the next shot he lies half on and half on the grass on his front.
Continuity mistake: In 'Consequences' Buffy says to Faith 'Less that 24 hours ago you killed a man'. In 'Bad Girls', Faith kills Finch in the alley on the first night. One night later Buffy and Angel are attacking Belthazor, the night after that is the night when Buffy dreams about drowning, so the night when Buffy says this is actually about 72 hours after the murder.
Audio problem: In the episode 'Once More, With Feeling', when Anya sings her first few lines of the song 'I'll Never Tell', you can still hear Xander talking about omelettes in the background, but for the last few lines he speaks (from 'I could do an omelette' onwards) you can see over her shoulder that his voice is dubbed.
Plot hole: Healing Angel requires draining the blood of a Slayer. Buffy tries to kill Faith in order to get some of her blood but Faith gets away after Buffy stabs her at the end of "Graduation Day Part 1." The amount of blood on the dagger that she used wouldn't be enough blood to heal Angel, but it'd be a start. So why didn't Buffy just use some of the blood on the dagger? In the very first shot of "Graduation Day Part 2" we see the dagger coated with the stuff.