Billing Displacement: Tobin Bell is always top-billed in the later sequels, despite being dead in the "present" and only appearing in a handful of flashbacks and tapes. The displacement is taken to ridiculous lengths in the final film, where Bell is top-billed in spite of being in the film for TWO MINUTES; Cary Elwes appears in three scenes, has four lines of dialogue, and yet he is billed above Sean Patrick Flanery, who, at fifth billing, has the most screentime of anyone in the film and is essentially the main "hero" character; Chad Donella, who appears throughout the film and is the lead cop in the movie, doesn't even make the top five or the posters; Costas Mandylor and Betsy Russell get second and third billing, respectively, even though the former is barely seen until the end, and the latter, while having more screentime that Mandylor, doesn't do a heck of a lot - their billing can be pinned on their being staples of the series, having appeared in every film since Saw III, rather than their roles in the plot.
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.
Mean Character, Nice Actor: If you watch the behind the scenes featurette, Frankie G (who played the murderous drug dealer Xavier in Saw II) is a really nice guy.
The first film underwent some significant changes in trap design and deaths.
Mark (who dies from the flammable jelly) was supposed to die from poison after using his arm for a torch.
Paul, 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).
Jeff, the victim of the Drill Trap, dies in the script but is saved in the finished movie.
According to the commentary and different interviews, the creators wanted a trap that consisted of two things that would spring from the walls and snap shut on Sing and fold him up into a box. They described it as an iron cocoon. They decided against the idea, as it would have cost too much money for visual effects. A similar idea would later be brought up for the original concept of The Rack, which would again be scrapped. This trap was later used in the video game as a trap for Sing's wife, ironically enough.
According to an interview with Leigh Whannell, Adam and Lawrence were originally going to be trapped in an elevator or a basement instead of the Bathroom.
Eric Matthews' name was originally Eric Mason.
According to the commentary, a blade had originally been intended to shoot forward out of the staircase and slice off the first SWAT officer's legs once said rigged step had been activated.
The creators and writers had originally thought of having Obi's tape recorder attached to the collar of a pit-bull in the basement. They decided against it, as it would lead to cruelty to animals, and instead attached it to a corpse-like mannequin.
It was originally intended for Amanda to land up to her neck in the needles as if it were a pool, but realized that it would be impossible for her to sink into such a pool. Also, getting enough needles would have taken too long, as it took the production team a long time to get as many needles as they had.
Addison was meant to press her face against a waffle iron. This trap was later redesigned as the knife chair in Saw IV.
According to the DVD commentary for Saw VII, Lawrence Gordon was meant to be the main character, trying to save his wife from Jigsaw and Amanda and eventually killing them. This ended up being Jeff's role in the film itself. It is unknown whether or not Elwes was asked to return for the film or if he declined in the aftermath of his lawsuit following the first film.
The scene showing Amanda stabbing Eric Matthews in the throat with a knife was filmed, but it was cut so that he could be alive for Saw IV.
Several trap redesigns.
One idea had Troy suspended above the floor of the classroom by large meat hooks. They decided against this, as they figured it would be basically impossible to escape, and that it would be more thrilling if he was standing and able to rip the chains out. A second version had the hooks going through his eyelids, his fingernails, and his teeth. This accounts for two of the promotional posters for the film, one showing a mouth missing numerous teeth, and another showing three teeth dangling from wires. The teeth-pulling aspect was recycled for Bobby Dagen's test in Saw VII.
According to producers Mark Burg and Oren Koules, they had originally discussed having Kerry placed in a device that would pull her arms and legs in four different directions, but instead decided on the ribcage trap. However the idea was used in the Saw IV FEEL WHAT I FEEL trap.
Originally, the victim in the freezer trap was going to be a male police officer, but they decided against it, instead going with the idea of a female witness. Instead of being completely naked, Danica was going to be in a T-shirt and nickers, but they figured having her being sprayed would make the shirt stick to her body and become too sexual, so they decided to put her in the freezer naked instead.
One idea had Judge Halden suspended above a meat grinder. As he was lowered down into it, Jeff would have to dive into a "soup" (probably referring to the ground up pigs that were instead poured onto the Judge) to retrieve the key. It was changed again until it resembled the final version, with the exception that Jeff wouldn't have to burn his son's possessions, but simply step down into the silo to free the judge. Leigh Whannell said on the commentary that one of the reasons for the final change was that merely having Jeff dive into pig guts was more akin to something on Fear Factor, and that by burning his son's possessions, it was more of an emotional conflict for Jeff than a physical one.
Timothy Young's trap went through redesigning. One idea had Timothy strapped down to a table that would fold on its hinges, bending his body with it and breaking his back. A similar idea had been brought up once before when thinking about Jeff Ridenhour's trap. It was remodeled into the shape of a cross, but would still bend along hinges that would break Timothy's body until they decided they wanted the individual limbs being twisted instead.
Originally, Lynn's collar was going to be more along the lines of a bomb, hence Jigsaw's line, "an explosion will go off in that collar". The collar would detonate, shaking the room, knocking the dead Jigsaw from his bed, and knocking Jeff unconscious.
Originally Saw IV was supposed to be a direct sequel to Saw III, starting with Jeff and Chris dying in the blender trap. Eric Matthews was supposed to be the main character, passing his test with help of poisoned Hoffman, who at the end was supposed to be revealed as Jigsaw's apprentice.
It was suggested that the tricycle and Obi's connection to Jigsaw would be explained, but this never came to pass.
In regards to the mausoleum trap, the winch was originally going to be on the ceiling and eventually hang Art and Trevor. The final shooting script has Trevor tearing an eye open, and he dies by the chains pulling him too close to the machine, snapping his neck, rather than by Art killing him. Art is shown finding the tape recorder right after the trap, rather than waiting till the end of the movie to show it as a flashback.
Details from the final shooting script may be found here: 
Rigg tells Hoffman that he wouldn't go home until he found Eric, saying "I made that promise to Daniel." Sadly, Daniel Matthews was only referenced via reused photo in the film.
Speaking of photos, the final shooting script mentions pictures hung up in the police precinct of Tapp and Sing, with the message "Your sacrifice will never be forgotten."
Hoffman's trap has him hanging on a meat hook, with the pool of water collecting under him. The idea was that once time ran out, he would be released from the hook, and fall into the water, getting electrocuted.
The classroom for the Rex and Morgan trap is the same one from Troy's trap, and its significance in Rigg finding it is in the notion of "Become the teacher and teach her how to save a life."
In Jigsaw's first lair, when he shows it to Jill, he had miniature versions of all his future traps, all having been performed on rats, with mangled rats all over the room. This was probably cut so that he would be more empathetic.
Eric still thinks Amanda is being everything, because Art is initially walking around the room in a hood. Eric asks Hoffman if the "junkie bitch" got him too, and keeps yelling at the hooded Art as if it was Amanda. "You fucking bitch," etc.
At one point when Eric jumps off the ice block, Art comes over to him and Eric swings his legs around Arts neck, trying to break it, and the fight for a little while.
Jigsaw was setting up a TV in front of Cecil, and Cecil woke up early, so then John just played a remote that turned the TV on, and there was a doll message for Cecil, without the whole conversation that him and John actually had. Cecil's chair actually breaks, to John's surprise.
In the motel, with Ivan and the dog, was a homeless man sleeping on the floor called Vagrant. In the script, when Jigsaw takes the cover off the glass coffin, he is this man who is in the coffin. You can see his picture by looking up the cast on IMDB.
Darren Lynn Bousman, the director, had a different ending and fate for Rigg in mind that was never shot: "What if he didn't open the door? Again, make the twist being something different. He actually didn't do it. He stops at the door, he breaks down crying, doesn't save him. He gets up and sees Agent Strahm with the gun. Strahm sees Lyriq with the gun, thinks he’s Jigsaw and just shoots and kills Lyriq there."
Writer Patrick Melton wanted Rigg to join Hoffman in carrying on Jigsaw's legacy at the end, but Executive Meddling prevented this.
It was considered to have Detective Tapp appear in new footage via flashback, but Danny Glover was unavailable at the time.
Originally, the blades would be in the collars, instead of mounted on the wall in the five victims' first room. It was also originally conceived that both Brit and Luba would somehow lose their shirts in the first room and go through the rest of the film in just their bras. The idea was eventually thrown out, out of respect the producers had for the two actresses.
Originally, the final room was supposed to fill up with water, ending the scene by drowning Strahm. But the crew realized that it would be too difficult to shoot, so they re-thought the scene and came up with the finished version. The idea to have Strahm crushed between two walls actually came from director David Hackl's son.
Editor Kevin Greutert explained what was cut from the ending: "Originally, the "game of five" was connected more directly to Strahm. It's something they're trying to figure out all along. This is discovered in the final seconds and everyone believes that Strahm is Jigsaw's apprentice, essentially letting Hoffman off the hook. But it was much more fleshed out. It was also supposed to reveal that Jill knew that Hoffman was continuing Jigsaw’s work."
Back when the film was greenlit, it was announced that Scott Patterson would play Strahm up to this film. An old idea involved resolution of Strahm-Hoffman struggle and was supposed to wrap the series.
Lionsgate wanted Hoffman to fight mafia.
Director Kevin Greutert wanted Lawrence Gordon to be back seeking revenge against Jigsaw's accomplices. Survivors were also supposed to return, though in a different role.
According to an audio commentary with director Kevin Greutert, and writers Patrick Melton and Marcus Dunstan, Simone would originally cut off her fingers, then her hand, then her arm.
The commentary also reveals that there were supposed to be ten people riding the carousel (they changed it to six seeing as it is the sixth Saw film). Also, the shotgun would be placed behind the victims heads, rather than in front of their chests.
Brad thrusts the saw into Ryan, impaling him. Dina and Brad then survive the trap, but the script was changed to make Dina the loser and Brad and Ryan the winners of the trap.
Originally, Evan, the racist skinhead played by Chester Bennington, was meant to live through the Horsepower Trap. But because Bennington was under a very tight schedule at the time, and only had the one day for shooting, the filmmakers didn't have time to shoot his hospital scene, and Evan was rewritten to having died.
Daniel Matthews was meant to appear at the survivors' meeting, but Erik Knudsen was unavailable. Brent also did not appear in the scene due to Devon Bostick's scheduling conflicts.
Saw VII was originally meant to end with a Sequel Hook: the revelation that Lawrence Gordon was still alive. Saw VIII would have then explained how he survived. Due to the disappointing box office performance of Saw VI, the ideas for VII and VIII were condensed into one film. According to Patrick Melton, the original pitch was not to have Gordon trap Hoffman in the bathroom.
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.
Dr. Lawrence Gordon was originally supposed to appear in the first video game to act as Mr. Exposition.
Word of God: According to the writers' commentary track on the Saw: The Final Chapter DVD, the two men with Dr. Gordon when he captures Hoffman at the end of the film are Brad and Ryan, the two surviving parties of the trap that started the film.