On one hand, how would Sauron know about Durin's Bane? The dwarves didn't know what sort of creature they unleashed and it's doubtful that any orcs would be able to identify it, beyond it being evil and far more powerful that the lot of them.
Also, if Sauron knew about the Balrog and the Balrog recognized Sauron as a higher up on Team Morgoth, he's have almost certainly put the old demon to work doing something useful, like kicking over Galadriels house.
On the other hand, it is rather odd that the Balrog permitted Sauron's orcs to live in Moria unless he thought that Sauron was the boss. After all, Sauron was Morgoth's second in command, even back before most of the Balrogs were exterminated. The Balrog may have remembered this and thus didn't go on a orc kicking spree.
Also Sauron promised Moria to the dwarves had they joined him. It would have been impossible to turn Moria over to the dwarves unless the Balrog was willing to relocate. This, however, is a very weak argument, since Sauron may well have been giving them a false promise or even that he knew full well what was waiting for the Dwarves if they returned to Moria. The Dwarves take the bait, help Sauron win the war, go back to Moria, and whoops! The Balrog evicts them. Balrogs can't possibly make good landlords.

So what do you all think?