(at around 1h 8 mins) John Kramer sits a sawed-off shotgun onto two brackets shaped similar to tuning forks and they are at different heights. A few minutes prior to this, he removes a stack of saw blades from the brackets and they are both at the same height.
(at around 50 secs) During the car chase, the car runs over a harrow set by the police, but in the next shot, the front right tire is not flat.
(at around 13 mins) When Logan plays a hard drive found in the body of a victim, you hear the audio play, but when it cuts to the computer, it shows the audio file has just started playing.
When Mitch is being lowered into the "Spiralizer" the motorcycle powering it is running the machine with its rear sprocket. When it cuts to Anna trying to save Mitch by climbing up next to the bike, you can see the sprocket is not moving on the bike. Between cuts it alternates between rotating and not rotating.
(at around 30 mins) When Logan mentions that hydrofluoric acid was used to kill the second victim, he mentions that the tissue is connected to the heart, therefore everything, including the heart, is eaten away by the hydrofluoric acid. This is incorrect as hydrofluoric acid only eats oxides, and not organic tissue like flesh. If dropped on the skin, nothing would happen. It would seep into the body and begin dissolving the bones of the victim by a process known as decalcification.
(at around 31 mins) Detective Hunt says that Logan was captured in Fallujah after taking out three Talibans, which is impossible because Fallujah is in Iraq, while Talibans are in Afghanistan.
(at around 1h 2 mins) The sticker on John Kramer's X-Ray doesn't have his birthdate on it. All labels (on any item) for a hospital patient would have their name and birthdate to ensure accurate cross-verifying of identification.
(at around 34 mins) Since most of Jigsaw's voice is pre-recorded on tape and on TV, it would be hard to know the victim's name if they accidentally fall into the wrong trap. A little while after Ryan get his leg stuck in the Leg Wires underneath the floorboard two televisions switch on (not his trap or test). Jigsaw mentions Ryan's name. Since the Leg Wire Trap was made for anyone who tried to go through the exit, he would have no idea whose name it was. Based on the later flashback showing John rushing in to take Logan out of the bucket himself, it's pretty likely that he was watching the game live and could say the correct person's name, since the TV screen is showing live events, not a pre-recorded message.
(at around 49 mins) Just before Mitch starts to lower into "The Spiralizer" the close up of the tape player doesn't have moving numbers on the tape counter, even though you would see them progressing in the time frame.
(at around 47 mins) Detective Halloran looks up Logan on the Department of Veteran Affairs website and is able to pull up Logan's medical records. It lists his diagnosis and prescription history. This is all information that is protected by doctor/patient privilege,. It wouldn't be posted online for anyone to see. Even if Detective Halloran was accessing a database only available to law enforcement, he still would not be able to access Logan's medical history without Logan's written permission or a court order.
John Kramer states that Anna framed her husband for the death of their baby. But John cannot know this to be true. He overheard the couple arguing, but there is no possible way for him to know that Anna's husband did not roll over and suffocate their baby. Similarly, John claims that Ryan lied to the police regarding the car crash that killed his best friend. Unless John was a first-hand witness to this event, he could not possibly know the truth.
(at around 20 mins) When the chains begin to retract in Carly's trap, the case holding the three syringes gets lifted into the ceiling. If Ryan had not taken all three syringes out of the case before that point, there would have been no chance for any of the participants to survive. This goes against John's mantra of the participants making a choice to survive.
After Anna, Carly, Mitch, and Ryan make it through the "Bucket Room"; the main part of Tuck's Pig Farm looks like it's almost entirely made of wood. If the walls were made of wood, they could have found a way out considering that there are enough tools around.
Defective Bunnyreferences that Logan was captured by the Taliban in Fallujah. The Taliban are active in Afghanistan, Fallujah is in Iraq.