The second-year pro from Tennessee was injured in Arizona's 23-7 loss at San Francisco on Sunday. Williams was Arizona's first-round draft pick, the 26th selection overall, in 2010.
Coach Ken Whisenhunt says rookie David Carter and veteran Nick Eason will move into Williams' spot. Carter is a sixth-round draft pick out of UCLA. Eason, in his ninth NFL season, is a defensive end but has played some at the nose position.
On Tuesday, the team elevated defensive end Ronald Talley from the practice squad to take the roster spot vacated by Williams.
Whisenhunt also says starting right tackle Brandon Keith has a concussion and will be evaluated through the week to determine if he can play in next Sunday's game at St. Louis.
Kolb pushed his injured foot in practice last week but was "not very" close to being able to play, Whisenhunt said.
"I pushed it hard all week and it just wasn't there," he said. "I think I would be hurting my team if I went out there and tried to play on it. I'd be limping around out there and definitely wouldn't be up to par."
Kolb's injury is a turf toe along with a bruise on the side of the foot he uses to plant when he throws.
"It is frustrating," he said. "It's frustrating for everybody. The only injuries I've ever been through are pretty self-explanatory, a broken bone and a concussion. I've never been through anything like this before. But I'm going to keep pushing and do my best to get out there this week."
The offense also has been hurt by the nagging sore knee that has hampered Beanie Wells, although the extent of that seemed to be a subject of dispute between Whisenhunt and the running back.
Whisenhunt said Wells' time in the game was limited.
"He can't go more than a certain number of reps at a time during the course of a game, so you have to make sure that you watch that, then he comes out," Whisenhunt said. "There have been a few times where we've had plays called that we wanted him in there that you've had to put another back in there.
"But once again, you understand that's going on because of where Beanie is health-wise. To have him available is what's important going forward."
Wells, who carried eight times for 33 yards against the 49ers, said that was news to him. He said he's been limited in practice but ready to go full-tilt on Sundays.
"I don't think he was limiting my carries because to me that's something to worry about in practice to get to the game," Wells said. "We take it easy during the week to get me to the game, and once the game is here I'm full go, ready to play 100 percent of the plays."
Information from The Associated Press contributed to this report.