The receiver will be out four to six weeks and will be sent home from training camp in Phoenix to Los Angeles for further diagnosis and treatment, Torre told MLB.com Sunday.
The skipper said Dodgers rookie A.J. Ellis would "get the bulk of the playing time here," with 40-year-old backup catcher Brad Ausmus remaining in the mix.
"Ausmus brings a lot to the table with his experience, but I certainly don't want to run him into the ground either," Torre told MLB.com, adding he isn't looking to swing a deal with another team for a replacement for Martin.
Martin is a two-time All-Star who won a Silver Slugger award for the league's highest batting average as well as a Gold Glove in 2007, but his production has decreased since. He hit only .250 with seven homers and 53 RBIs last season.