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.
Continuity mistake: In the very last episode, Chosen, in the shot right before Spike's amulet sends out the blast through the roof, you can see an ubervamp just about to pin him against the wall and in the next shot he is standing in the middle of the area backing up toward the stairs and then he sends out the blast. There would never be enough time between the shots for Spike to break free and then go and stand in the middle of the floor.
Plot hole: When Buffy and Sheila are walking out of school after the meeting with Snyder, Xander runs up to talk to them. He mentions during this scene how Buffy will be helping out at the Parent-Teacher Night, but he shouldn't know this - Buffy only just found out about it in the meeting with Snyder.
Revealing mistake: The knives Giles throws during Buffy's training are a very poor CGI job if you watch it in slow mo. All three knives Giles takes off the wall are identical, with a broad curved blade and a plain black handle. The first one has a transparent blade as it flies toward Buffy, then we hear it bounce off the far wall and it's way out of scale for the distance. The second knife has a diamond-patterned grip instead of a plain one. The third knife conveniently spins in midair for a second so Buffy can kick it out of the way.
Other mistake: We're shown a variety of photographs and pictures of different members of the gang at one point. One of those, the one of Xander and Anya, is actually a shot from the episode "Something Blue". I doubt very much that during the events in that episode, someone conveniently decided to take a photograph.