Deliberate mistake: When Buffy and Giles are searching the funeral home, we see three shots of Buffy and Giles opening up different mortuary-cabinet doors, shot from the point of view of the cabinet. The problem? You can tell they're opening and closing the same door over and over again. The background is exactly the same, and all they've done is slightly re-arrange the stuff behind them, and had the actors come in from different angles to create the illusion they're opening different doors. It's SUPER obvious.
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.
Deliberate mistake: After Jenny pushes the janitor's cart into Angelus and runs up the stairs, there's no physical way he could be waiting for her at the top of the stairs (even with vampiric speed). Joss Whedon has admitted in interviews that he put in this blatant continuity error as the situation is a staple of the horror genre (girl runs from villain, only to find him standing right in front of her).
Deliberate mistake: Though vampires are said to possess super-human strength, and are in fact demonstrated to have it, they are often kept confined by the flimsiest of restraints. One distinct example is in season 2 episode 9 when Kendra locks up Angel he is unable to break out despite being held in by nothing but a flimsy padlock. This mistake is pointed out in the DVD commentary as an unavoidable 'mistake'.
Deliberate mistake: In the fighting scene, the glove of Myhnegon is used by the new watcher who directs it toward the ceiling of Angel's living quarters where there is a huge skylight. The skylight is not seen in any other episode and its presence would have burned him on other occasions. (00:38:10)
Deliberate mistake: When Xander ducks under the caution tape to catch up to the group of students at the beginning, as the shot is about to cut, judging by his casual walking pace, he should maybe be 10 or so feet ahead of Buffy and Willow. When the camera cuts, he's suddenly at least 30 feet away. It's probably edited that way for time reasons (to quicken the pace of the scene rather than show him walking an extra 10 seconds), but still a minor mistake.
Deliberate mistake: When Willow goes into her dark bedroom, she turns on her lamp. Somehow, her tiny lamp illuminates the entire room and also casts a shadow behind her, even though it's behind her. Clearly, when she turns on the lamp, they also turned on some set-lights in front of her to make the room bright enough for filming. There's also something else that gives it away-when she turns on the lamp, the studio lights come on a split-second later - just enough that if you look closely, you'll notice the delay.
Deliberate mistake: When Buffy, Xander and Giles try to "delete" Moloch from the computer, watch the computer screen as she sits down. There's already a folder labeled "Willow" on the desktop. Buffy then types in the word "Willow" to search for the document (without clicking on any sort of search bar, so typing it would do nothing), then says something along the lines of "That's probably it" and proceeds to click on the "Willow" folder... that was already on the desktop. The entire scene makes no sense.