This isn't good news for the Pirates, who had Pittsburgh pretty excited with the addition of Burnett. However, I don't think this is going to cost Burnett more than a couple starts, if any. He should be able to throw bullpen during spring training even just days after the surgery, although it will almost certainly hold him out of a good amount of spring games.
The biggest concern with pitchers in spring training is getting their arms stretched out to get ready for the long season. I don't think this injury is going to hold Burnett back too much from that. It also helps that Burnett has been around for a long time, so I'm sure he knows how to handle something like this.
The Pirates will release the timetable for Burnett's return tomorrow, so keep an eye out for that.