I have stripped a triple fireplace with shared chimney in the center of the house down to the brick masonry.  I am planning to glue or nail fireproof cement backer boards onto furring strips attached to the chimney (with firestops between the strips).  The boards will then be painted, or possibly tiled/stoned over if I don't like the painted look.  Any safety concerns I should be aware of?  Also, is there any reason to insulate the air space between the furring strips?  If so, with what material?

Someone else may probably have a more informed answer, but I would think you would want the thermal mass of the brick not to be insulated so that it could radiate the heat of the fireplace into the living space.

Agreed, I wouldn't insulate. Tile will stck to unpainted backer board better than painted.

I'm not exactly sure what you stripped away, however, every code I am aware of stipulates min of 2" clearance to combustibles, so attaching furring strips to the remaining mass may create some issues.

One issue in particular, if there are combustibles placed where they do not belong, and for some reason you have a fire, I doubt your insurance will give you more than the time of day!

Why not check with your local building inspector for a little input?

The house was built in the 60's when the code was less strict. There were furring strips, studs and plywood attached directly to the chimney with nails, though because of the massive size of the chimney (10'x6') there probably was no real risk of fire. The code does require a min 2-3" of clearance for combustibles, so I ended up cementing wonderboard directly onto the chimney, then finishing over with plaster. Thank you for your assistance.