The Clarets have lost yet another player for an extended period after Rodriguez suffered a fractured ankle.
The local-born 20-year-old striker becomes the third of Owen Coyle's players to be ruled out until Christmas in the last fortnight, with Martin Paterson and Chris McCann also both on the sidelines with knee injuries.
Fernando Guerrero tweaked his hamstring and Kevin McDonald sustained a knee problem in a midweek reserve game, leaving both looking doubtful for the visit of Alex McLeish's Blues to Turf Moor.
But club captain Steven Caldwell is in contention for a place in the starting line-up after playing for an hour in the same match.
Burnley (from): Jensen, Caldwell, Mears, Eckersley, Duff, Carlisle, Bikey, Jordan, Alexander, Fletcher, Elliott, Blake, Thompson, Guerrero, Eagles, Penny, Kalvanes, McDonald, Gudjonsson, Edgar, Easton, Nugent.