Visible crew/equipment: After Tara is found in the Janitor's closet and brought back to her room as Willow & Buffy come in, there is a mirror to the left of the door. It has a scarf draped over it to prevent it from reflecting things it shouldn't, but the bottom right corner is still exposed and in it you can see glimpses of the crew moving around behind the camera.
Visible crew/equipment: While the house is becoming "Wild", Riley takes Graham down to The Initiative. Once the door closes behind them and the mirror returns to its original position, the shadow of a camera is reflected in the mirror. It's in the middle of the frame, on the left, over a book case.
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.
Continuity mistake: In the scene near the end when Faith is putting Joyce's lipstick on in the mirror and taunting her, it is obvious that the make-up department forgot to put Faith's tattoo on her upper arm. It's obvious if you watch back and forth between the mirror and the shot of Faith standing and turning, there is no evidence of a tattoo that has been established Faith has in the previous season.
Other mistake: Just after Giles becomes completely blind, Spike gets up to get a book off of the mantle. When he does this you can, very briefly, see James Marsters reflected in the glass doors of the bookcase that was behind him, despite Spike being a vampire. (00:27:00)
Continuity mistake: When Spike finally gets into the area where the gem is and Harmony follows him up there, he puts on that necklace thinking it's the gem. When he realizes Harmony has the gem it shows him taking off the necklace. It cuts to another shot and then shows him again with the necklace, but minus the charm.