Visible crew/equipment: When Buffy and Xander are doing some research, check the left side of the screen in a wide shot, when Xander asks "how far back do the disappearances go?" You may need to turn the brightness up, but there's a boom mike bobbing around.
Revealing mistake: Early in the final fight, Vi makes a flying kick to knock an Ubervamp off the cliff. As she lands, she swings back and twists slightly, suspiciously like one would do if suspended on a wire. There is no way she could gain this motion by kicking straight off the vamp as shown.
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.
Continuity mistake: In 'The Initiative', Riley, Graham and Forrest are looking in through the windows of Stevenson Hall with thermal goggles looking for Spike, they see him walking towards Willow. When we see Willow and Spike seconds later he is still sitting on the bed where he was for most of the scene.
Continuity mistake: In numerous episodes, including 'Homecoming', we see vampires being killed when they weren't stabbed through the heart (the Texan girl vampire in this episode is stabbed practically in the stomach). But the show also makes an issue of the fact you have to stab a vampire directly in the heart to kill it (think the flashback in 'the Becoming, Part 2' where Buffy misses the vampire's heart, or 'Pangs' where Spike gets staked several times).