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.
Continuity mistake: Towards the end of the episode whilst Buffy is fighting Caleb with the scythe, right after he says to her, "You're not slipping out of this fight, girl. Can't you see? You can't stop me. I can just keep going back for more. It's like being reborn," and throws her into a wall, the position of the scythe changes constantly between shots.
Continuity mistake: Sunnydale apparently has docks ('Consequences' and 'Surprise') and is quite close to a beach ('Go Fish'), but when Sunnydale is destroyed in 'Chosen', we see an overhead shot of the crater and it appears to be in the middle of the desert, far from any coastline.
Visible crew/equipment: After Tara is found in the Janitor's closet and brought back to her room as Willow & Buffy come in, there is a mirror to the left of the door. It has a scarf draped over it to prevent it from reflecting things it shouldn't, but the bottom right corner is still exposed and in it you can see glimpses of the crew moving around behind the camera.
Factual error: After Buffy drags Xander down to the basement she locks the door and goes upstairs to find Dawn. When she locks the basement door, the handle is on the left side of the door and she turns the key to the right. In reality, a door with the handle on the left needs the key to be turned to the left to lock it. So Buffy did in fact unlock the door, not lock it.
Visible crew/equipment: When Buffy, Willow, Xander and Giles all agree that Morgan is their one suspect, a camera and a cameraman can be seen in the reflection of the glass in the door labeled "Librarian", right after Buffy says, "Uh, priority check, Giles." A boom mic is also reflected in the top right of the glass when Buffy finishes her line with "talent show murder!" The mic can be seen throughout the rest of the scene from two different camera angles.
Other mistake: In the famous scene where Buffy tells Giles, "If the Apocalypse comes, beep me," she grabs her pager from out of shot. Trouble is, she's standing in the middle of her hall and there's nowhere she could have picked the gadget up from.