In Saw II, Amanda Young slit her wrists with razor blades. By Saw III the cuts would have healed but left marks or scars. There are no visible marks or scars on her wrists but in Saw VI, the flashback sequence which took place during the events of Saw III clearly make a reference to the scars on her wrists.
(at around 1h 2 mins) When Lynn is operating on John's head, she cuts the skin from the skull and then clips off some veins. Right when she asks Amanda to fill the syringe with alcohol to clean John's head, Lynn turns back to wipe up some of the blood, but there are no clips in place. A shot after that, we see her taking the syringe from Amanda's head and squirting the alcohol on John's head and the clips are there.
(at around 1h 3 mins) When Lynn, the Doctor, is drilling the four holes in Jigsaw's head for the operation, tape is on the drill to mark the maximum depth. The placement of the tape changes positions throughout the scene, between 1/2 inch and 1 1/2 inches.
While Kerry is in the angel trap, and the key is first dropped in the acid it automatically begins to bubble immensely. But while the tape plays explaining the game and her objective the key does not bubble there is nothing happening in the jar but immediately after he says in a matter of seconds it bubbles profusely again once the camera focuses on the jar.
(at around 1h 11 mins) In the original Saw, Cary Elwes's character Dr. Gordon is very clearly blond. However, in the bathroom preparation scene in this film, Elwes's stand-in has very dark hair. Though it is tempting to blame this on the extreme contrast of the shot, Jigsaw's beard still registers as blond.
(at around 49 mins) The picture Jigsaw shows of Amanda in the flashback before she is captured and placed in her trap is actually a photo of her from Saw, where she is interrogated about Jigsaw and explains what happened to her *after* she was in the trap.
(at around 18 mins) When Lynn treats the child's pneumothorax with a drainage, the nurse exclaims that the patient is improving. However, at that moment the monitor shows a ventricular fibrillation (VF), a severe arrhythmia that causes death unless immediately treated. Moreover, the VF spontaneously turns into a stable "normal-beat" heart rhythm without an electrical shock, situation that's almost impossible to see in the clinical practice.
(at around 1h 40 mins) When Jigsaw pulls out the tape player at the end of the movie, his thumb is on top of the tape player, nowhere near the play button on the side of it. However, we distinctly hear a click (as if the play button had just been pressed), and the tape begins to play, even though the play button was never pressed.
When Lynn is wearing the "Shotgun Collar," close-up/insert shots show the primers on the shotgun shells already discharged. This means the shells had already been fired and would not work. While we could imagine that empty shells were used as containers for explosives, this is inconsistent with the collar's design (hammers are set to strike each cartridge) and effect (the collar, cartridges, and most of the victim above the chin are intact after firing).
The goof items below may give away important plot points.
(at around 1h 30 mins) When Amanda fires her gun in the finale (and the repeat shot in the flashback montage), it sounds as if it's silenced, when indeed it isn't. Ironically, this error is rectified when the event is repeated in Saw IV.
(at around 1h 35 mins) When Amanda is shot in the neck by Jeff, she falls to the floor next to a white cabinet. As the scene continues, the cabinet switches back and forth between having been splashed by her blood, and being perfectly clean.
(at around 15 mins) When Kerri is in the "angel" trap, before she plummets her hand in the acid, you see her hover her hand above it for a few seconds, but you see her actually touch it just before the cut... yet she doesn't even flinch.
(at around 14 mins) In the "angel" trap, Kerri is told she has to reach in and get the key out of the acid. Right after the tape finished playing, we see Kerri struggle with the chains holding the beaker for a second and then reach into it. This shows us that because of the way the beaker is suspended and the way she is restrained, she is unable to tip the beaker far enough to spill any of the acid.