The Balrogs were fire-themed Maiar servants of Morgoth Bauglir. The Orcs, problematic though their origins are, were likewise servants of Morgoth.
For of the Maiar, many were drawn to his splendour in the days of his greatness, and remained in that allegiance down into darkness...
And in Utumno he gathered his demons about him, those spirits who first adhered to him in the days of his splendour, and became most like him in his corruption: their hearts were of fire, but they were cloaked in darkness, and terror went before them: they had whips of flame. Balrogs they were named...
—J. R. R. Tolkien, The Silmarillion
Seems reasonable that the Balrog would employ the Orcs, or at least not make wiping them out its agenda. Whether or not the Balrog of Moria would cooperate with Sauron in any meaningful sense, because both Sauron and the Balrog of Moria were servants of Morgorth, it seems reasonable that they might share broad values... as in Orcs make useful servants or tools, and Dwarves are enemies.
Part of the problematic origin of the Orcs is the question of free will: if they do not have free will, they can't really be evil, any more than a tornado or pneumonia is evil. But if they do have free will, then they can choose not to be evil... something that (a) never happens in the text, and (b) makes morally repugnant the kill-Orcs-on-sight policies and preferences of all the protagonists of The Hobbit and Lord of the Rings. The Balrog (and Balrogs in general) were not, however, creations (or corruptions) of Morgoth, but lesser angelic spirits (i.e. of the same general order as Sauron, Olórin/Gandalf, Melian, etc.), so in as much as Maiar have free will, the Balrog of Moria is at liberty to make its own choice regarding the Orcs.