Continuity mistake: Willow is turned into a ghost. Throughout the entire episode, she can't touch anything (can't turn the pages of a book, etc.), yet towards the end when she and Giles go to Ethan's shop, and she leaves, you can hear that she opened and closed the door, even though she can walk through walls and can't actually touch the door. The curtain moves when she leaves too.
Plot hole: If Spike and his gang only take a few hours to break through the door to Ford's club in 'Lie To Me' (which we are told is six feet of solid steel), why can't Angel break through a thin wire door that visibly shakes every time he hits it, given the same amount of time?
Audio problem: When Xander, Willow and Ampata meet up at The Bronze, the song the band is playing is at the point of a guitar solo. If you watch Oz (who has already been established as 'lead guitarist') on stage in the background; he's blatantly NOT playing that solo. The hand movements don't match up.
Visible crew/equipment: After the history teacher writes, "Don't walk away from me, bitch!" on the chalk board, Buffy says to Xander, "I'm telling you, something weird is going on." As she says "something", a boom mic is reflected in the window behind her on the right.
Deliberate mistake: After Jenny pushes the janitor's cart into Angelus and runs up the stairs, there's no physical way he could be waiting for her at the top of the stairs (even with vampiric speed). Joss Whedon has admitted in interviews that he put in this blatant continuity error as the situation is a staple of the horror genre (girl runs from villain, only to find him standing right in front of her).
Other mistake: Joyce tells Cordelia to grab some bandages for Xander from the bathroom upstairs. Joyce and Xander are in the kitchen and there is a small corridor that leads to the stairs, but Cordelia goes around the other way leading to the living room. Fair enough, Cordelia made a mistake. She isn't that bright. But she returns with the bandages coming back from the living room. Why come downstairs and then go all the way around through the living room when she could have just walked down the small corridor?