Saw III was intended by the writers to be the swan song for the series. The producers wanted more sequels, and had the writers thus make changes to the film to leave room open for future installments. Many of these were fairly minor, such as a shot of Jigsaw pouring wax over a tape that later played into Saw IV. The biggest change, however, was that of Eric Matthews' death. In the original cut, Amanda was supposed to have stabbed him in the neck with a knife, solidifying her transition to a full-out murderer killing out of malice (she had previously killed Adam as a way of mercy-killing). The producers forced director Darren Lynn Bousman to cut it out to leave room for Matthews' return in a later sequel. Thus, the scene now ends abruptly with Amanda simply walking over towards Matthews, and an ADR line of Jigsaw later saying to Amanda that she had merely left him for dead was added. Keep in mind that Donnie Wahlberg (the actor playing Eric) only showed up for Saw III with the promise that he would be killed off, so he was pretty pissed off when they cut out his scene. Darren had to beg him to come back for Saw IV.
A massively cunning, yet very dickish move by Lionsgate and Twisted Pictures, happened two weeks before the shooting of Saw VII began - they contractually forced the director of Saw VI, Kevin Greutert, to return and direct Saw VII while having David Hackl, the original director, shifted over to a yet-unconfirmed Lionsgate movie. This was due to the fact that Greutert was originally set to direct the sequel to Paranormal Activity, which is scheduled to be released on the same date as Saw VII in a similar situation to how the original film contended with Saw VI in 2009. Needless to say, both Greutert and Hackl didn't take that move very well.
I Knew It: In Saw 3D, it's revealed that Dr. Gordon survived the first film and had been disciple of Jigsaw working behind the scenes since then. Some fans theorized this originally based on a brief part of the video shown to the key-behind-the-eye victim in Saw II. A cloaked figure with a limp (assumed later to be Jigsaw) is seen next to an operating table. Saw 3D reveals that this WAS in fact Dr. Gordon.
The first film underwent some significant changes in trap design and deaths. Among them, Mark (who dies from the flammable jelly) was supposed to die from poison after using his arm for a torch, the victim of the "Razor Wire Maze" was originally condemned for his greed (in the finished film, Jigsaw is punishing him for attempting suicide despite living a satisfying life), and the victim of the Drill Trap dies in the script but is saved in the finished movie.
Apparently, the original story in the first Saw game involved a one Dr. Gordon that you would encounter toward the end and get clues about throughout the duration of the game, but upon review by Lionsgate (after the game was nearly complete), they nixed the idea because they had plans for Dr. Gordon in the seventh Saw film. This caused the ending to be retroactively changed to instead involve Melissa, a character encountered earlier in the game, allowing already existing character artwork to be used rather than scrambling to create a brand new character at the last second.
It's also been revealed recently that the first Saw game had to be resubmitted to ratings boards several times, toning down the gore, until finally passing with an M (Mature) rating. In fact, if you look closely at the PAL version of the game for Europe, you'll notice the gory trap cut scenes are edited differently from the North American version. Europe was apparently a little more lenient with what the developers could get away with, but even that edit of the game was toned way down from the developer's original, much gorier, intentions.