Plot hole: Spoilers. It's revealed that the barn game takes place at an old farmhouse owned by the family of Jill Tuck - Jigsaw's widow. It's public knowledge that she was married to Jigsaw and that buildings they owned served as the headquarters of several past traps, so the barn should have been investigated at some point in the meantime. Ten years have passed. It makes no sense that the barn was never investigated and that the bodies of the barn victims were never discovered.
Revealing mistake: During the opening trap, at one point one of the five "Bucket Heads" starts banging on the wall behind them. If you look very closely, you can see the set-wall waver slightly as it's struck, revealing it's not really a concrete wall. (And it's not supposed to be the doorway that is later revealed during the final twist. It's a separate section of wall). (00:05:40)
Audio problem: When Halloran and Hunt are discussing orders from the commissioner to exhume Jigsaw, it's clear that several of Halloran's lines have been looped in post. His voice is a bit too loud and a bit more forceful than he appears to be speaking, and at least one word doesn't quite line up with his mouth movements. It can be hard to spot the first time, but if you look for it, it's noticeable. (Confirmed in DVD/Blu-Ray commentary - the actor lost his voice and his lines had to be looped).
Deliberate mistake: During his autopsy in Saw IV, it is revealed John Kramer was 52 at the time of his death. Kramer's role in this film is the very early stages of his time as Jigsaw, shortly after his cancer diagnosis, which would put him in his early 50's. However, he looks far too old to convincingly be that young (Tobin Bell was in his mid-70's at the time of filming). He even looks older than he did when he was on his deathbed in Saw III.
Other mistake: During the opening trap we see Mitch strike the blades with his chain and get yanked forward. However, despite being yanked forward he still manages to stand in line with the others which he shouldn't be able to do, as his chain was shorter due to the extra tug he received. (00:06:30)