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.
Audio problem: When everyone is in the Bronze - just before the vamps show up - you hear the first line of a song being played, Ballad For Dead Friends by Dashboard Prophets, cut to outside and you see the vamps strolling along and killing the doorman, then cut back into the Bronze and the same line plays again. This is a total error in cutting the show, it is stated there is no live band but still the same line plays twice when that is not how the song goes.
Add timetroy fox
Continuity mistake: When Buffy is talking to Owen at the end, there is a woman wearing a red/black checkered jacket, sometimes carrying a red bag, walking up the stairs behind Owen several times. You can also see her behind Buffy in shots facing her.
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.
Other mistake: In the episode 'Anne' Kenn tells Buffy that time in his hell dimension passes more quickly, that one day on Earth is the same as 100 years in his dimension. This means time goes roughly 36,000 times quicker. When Lily falls into the portal, it's at least 10 seconds until Buffy jumps in after her, but Lily is still just sitting there looking stunned. If times moves that much faster, 10 seconds would become over 4 days. Would Lily still have been just sitting there?