Sophomore Rhett Rodriguez, the son of former coach Rich Rodriguez, will start in his place.
"Going through the week, he felt pretty good," Sumlin said Monday. "But we can't keep going with this thing getting tweaked, and two series [later] he's out of the game."
Tate was pulled from Saturday's loss in a move that Sumlin called a "player safety issue."
"We're never going to put someone in harm's way," Sumlin said.
Tate has been dealing with an ankle issue since an injury vs. Houston on Sept. 8, and he has been largely ineffective after beginning the season as a Heisman Trophy favorite. Through seven games, Tate has rushed for 112 yards, which is glaringly low considering he rushed for 1,411 yards last season when he began the year as the backup quarerback.
It will be the first career start for Rodriguez, who completed 20-of-38 passes for 226 yards and a touchdown in last week's 42-10 loss to Utah.