Plot hole: Willow hacks into The Initiative's computer system and discovers air ducts and electrical conduits leading to Adam behind 314, but the room doesn't show up on the schematic. It seems unlikely that nobody who didn't know about Adam would have noticed this and investigated.
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: 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.
Continuity mistake: Buffy's birthdate starts at 1979. In "I Robot, You Jane", when the demon pulls up Buffy's file in the school system, it shows her year of birth as 1980 and her year as sophomore. But when her file comes up on the student's computer screen, it's changed to 1979 and that she is a senior. It wasn't until towards the end of season 1 in "Nightmares" that her birth year is 1981 (on her tombstone before she comes out as a vampire).
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?
Continuity mistake: In episode 'Same Place, Same Time', Gnarl cuts Dawn across the stomach paralysing her and ripping her shirt, but when Buffy and Xander are carying her home there is no tear in her shirt or any blood. Also when Buffy goes back to kill Gnarl he tears her shirt but seconds later the rip has gone.
Deliberate mistake: Every time the commandos are seen in the early episodes of this season (The Freshman and Fear, Itself) they wear ski masks to protect their identities. Starting in The Initiative, they stop doing this. The reason is that the audience needs to be able to identify the different commando characters, but it's still a discontinuity.