Revealing mistake: When Eric Matthews is smashing his foot, he hits it the first time and it shows a close up. If you watch the small amount of blood when he hits it again they have used the same shot.
Continuity mistake: The drill that Lynn uses on Jigsaw's skull has a white piece of tape on the end of the drill bit a short distance from the end, presumably to mark the thickness of his skull so she knows where to stop. In one shot (during the drilling scene) the piece of tape moves a good two inches farther up on the drill bit, but it's back down in the very next shot.
Continuity mistake: The tear in Jeff's face (after he gets stuck to the refrigeration machine) continually changes throughout the movie. It goes from oozing and bloody, to clean and almost normal looking several different times. There's no blood on his bathrobe, so he didn't wipe it off that way. That's a big enough tear to bleed a decent amount, so we'd see the blood if he wiped it off.
Continuity mistake: Right before Lynn says "I don't know what you know, but my marriage has survived more suffering that you could possibly imagine" her hair is all pulled back behind her ear. As she's saying that line you can now see that a noticeable amount is hanging on the side of her face and across her ear.
Continuity mistake: The glowing "X" on the bathroom wall in the first Saw movie was very symmetrical, and the whole thing was filled in very well. During the Saw III bathroom flashback scene, when Jigsaw paints the "X" it's completely different that the one that shows up in Saw. It's not symmetrical at all, and the paint doesn't go on very thick at all. You can easily still see the wall behind the paint, but in Saw, the lines of the "X" are very full with no gaps.
Continuity mistake: Jeff sees the 3rd tape player (in the room where he finds the judge) and there's a shot of the player being held up by string. The player is hanging very level and the string is perfectly tied around the middle of it. Jeff climbs the stairs to get the player and now you can see that the player is hanging crooked and the string (where it's tied) is slanted towards the top of the player. (He didn't shake it lose or anything when he climbed the steps, the player is hanging from the ceiling).
Continuity mistake: After Lynn gets shot her position against the wall changes between the far away and close-up shots. (look at the poster she's leaning against for a reference point, in the far away shot her head is between the 2 legs on the left, but in the close-up her head is on the very left leg and you can easily see the right leg.).
Continuity mistake: The placement of the flashlight (or foot) moves between shots of Detective Matthews contemplating sawing through his leg. In one shot the flashlight is illuminating the ball of his foot, in the next it's pointed past his foot, then in the next shot it's shining right on his heel. Matthews isn't moving either, he sitting right there with the saw on his leg, crying and contemplating all the pain that he's about to inflict on himself.
Other mistake: After Jeff gets the key from the Rack trap room, the shotgun fires and hits the judge. He is standing with the shotgun to his right, yet he is hit on the left side of his face. There is also an inconsistency with the timing. The gun fires, then the actor turns to reveal the wound on the left side of his face. It's easier (but in no way required) to see the glaring mistake with slow motion.
Continuity mistake: When Jeff leans down to Lynn you can see she she has some blood on her chin and neck. The shot goes to Jigsaw, then back to Lynn and Jeff. Now her neck and chin have no blood. Shot moves to Jigsaw again, then back to Jeff and Lynn. Now she has blood on her neck and chin *and* blood from the corner of her mouth.
Continuity mistake: When Lynn places down the drill there are 4 pieces of a broken mirror on the table in front of her. The 2 middle ones are standing straight up, but the outer ones are at various angles of slant. The next shot (a few seconds later) all 4 pieces are standing straight up and arranged very nicely.