Is it safe to burn stained wood in a fireplace?

NO! Thats Highly dangerous as varnished, painted, and green treated woods release terrible toxins as they burn. Residue from treated wood sticks to the chiminey liner, if enough is burned, a liquid form of the residue will acumulate and run down the chimney liner.

Never burn painted, stained, or treated wood or manufactured wood such as plywood and particle board. Chemicals in 'salt treated' wood, paint, or stains can produce toxic fumes when burned. Likewise, burning manufactured wood products produces toxins and carcinogens.