(at around 1 min) During the beginning autopsy scene, the person performing the autopsy puts his mask on upside down (the eye splash guard is at his neck). As soon as the camera jumps, the face guard is positioned correctly.
(at around 34 mins) When Officer Rigg enters the hotel room with the box on the bed, there is a photo of his wife on top of it. On the back of the photo, the words "she needed you" are clearly displayed in red ink. However, as he opens the box, you can see the that the back of the photo now reads "she needs you" instead from the letters "eds" at the edge of his hand.
When the scene with Perez on her way to the hospital ends, it shows Art sitting at the desk with the timer on 19.47. Later in that scene he walks back over to the desk and the timer is on 19.39 when about a minute of the scene had gone by.
(at around 46 mins) During the flashback in which Jill is attacked by the man who "forgot" his jacket as she is leaving the clinic, she has to unlock the door at knife-point even though she never re-locks it when retrieving the jacket. The door's lock is a deadbolt type and no knob is present, so it wouldn't lock itself automatically.
When Rigg finds Jigsaw's last note for him in the desk drawer of the "FEEL WHAT I FEEL" room, it is first seen to have only one sentence: "Become the teacher and save a life." In a subsequent close-up, a second sentence appears: "Go back to where it all began."
At the end of the camera pan from the hospital room into the interrogation room, Jill's shirt is not fully buttoned as the character had to run from the hospital set into the interrogation set while changing costumes. It is corrected in the next shot.
(at around 2 mins) In the beginning of the movie, during the autopsy, John's skull is seen having the minor rectangular piece removed. This, however, is not possible, as Lynn from Saw III removed the piece as part of his brain surgery.
When Agent Strahm has his outrage, after he has called Agent Perez's mother, the first time he knocks down the papers the computer screen shows "normal use". When they show the knocking over again zoomed in the computer screen shows the windows welcome screen, then back to "normal use".
(at around 3 mins) During the autopsy, when the brain is taken out of the skull, it is not attached to the brain stem and spinal cord. it is loose in the skull. in reality, lots of dissections are needed for it to be done.
(at around 15 mins) When Rigg is watching the interrogation video of Jigsaw's ex-wife he is looking at several books and magazines about Jigsaw. The book he picks up is titled "Jigsaw: Is He The Murderer The Police Say He HIS?"
(at around 1h 8 mins) In a flashback, John is shown abducting Cecil at a Chinese street festival celebrating the Year of the Pig. The last two years of the Pig were in 1995-1996 and 2007-2008. This does not fit well with Jill's statement that these events took place shortly after John's diagnosis with cancer in 2004.
(at around 3 mins) At the beginning of the movie, the organ scale in the background can be seen reading just under 1/2 pound. The needle never moves, even when John's brain is taken out of it a minute later, it still reads 1/2 pound and never moves to zero pounds when emptied.
The goof items below may give away important plot points.
Incorrectly regarded as goofs
At the end of Saw III, when John died, he has no facial growth, and during his autopsy, stubble is clearly visible. Despite it being commonly believed that a person's hair can grow after death, it cannot. However a corpse's skin can recede allowing hair under the skin which grew during life to become visible.
(at around 1h 18 mins) Art's second trap was the Spine Pincer, which was intended to slice through the back of his neck if he failed his test, but when he takes off his jacket (revealing the device) he bends his head and clearly shows that he can get his spine out of reach of the blades. The pincers have no extension, so Art really has no obligation and no reason to subject Eric and Hoffman to their torture.
When Agent Strahm and Agent Perez debut at the discovery of detective Kerry's body, Strahm declares his theory of a third person helping Jigsaw and Amanda Young. He could not possibly know that Amanda Young was involved at that stage.