McCoy suffered the concussion last week when he was hit in the head by Pittsburgh's James Harrison. After spending a short time on the sidelines, McCoy went back in the game.
Team officials said during the week McCoy was not given a sideline check for a concussion because no member of the training staff realized the severity of the hit.
Harrison has been suspended by the NFL for one game.
"Colt McCoy will not be traveling with us (to Arizona) so that means, of course, that Seneca Wallace will start the game," Cleveland Coach Pat Shurmur said. "Seneca will be backed up by Thad Lewis."
McCoy was sent home on multiple occasions this week due to headaches.
"I saw him this morning," Shurmur said. "He's still having some symptoms and that's why we sent him home. I wouldn't know exactly how to characterize it, but we'll talk about him, obviously, next week when we get back."
McCoy's father has been critical of the Browns this week, saying his son was clearly in need of medical attention after the hit by Harrison and should not have been allowed to go back into the contest.