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).
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).
Trivia: The film is edited by Kevin Greutert, who edited "Saw" 1-5, and directed the sixth and seventh films. Greutert is one of only a few holdover crewmembers from the original run of the series to return. He said he felt an obligation to be a part of "Jigsaw"- feeling that given his past with the series, he should help usher the franchise into the new direction that the producers and writers wanted to take, while also helping to maintain ties to what came before.