Burish was injured in Tuesday night's 6-0 loss to the Florida Panthers at American Airlines Center.
"He got hit with a puck," Stars coach Glen Gulutzan said. "Brenden (Morrow) shot the puck and it hit him in the hand."
Burish's injury comes on the heels of the Stars losing defenseman Alex Goligoski for about four weeks with a broken thumb. He was hit with a puck in Friday's game at Pittsburgh.
Gulutzan said forward Krys Barch is day to day after twisting his ankle in Tuesday's game.
The Stars do have some good news on the injury front. Steve Ott, who has missed the past six games with a hip injury, is expected to return for Friday's game at Colorado. Ott will take the place of Burish on a line with Morrow and Mike Ribeiro.
Mark Stepneski covers the Dallas Stars for ESPNDallas.com.