There are many structural concerns. Assuming this is a masonry chimney, its entire weight is born by the fireplace and then down to the footings beneath the fireplace. Demolishing just the fireplace basically leaves the chimney with no support. A collapse of the chimney - in who knows what direction - may or may not occur immediately, but the first high winds... In short, don't attempt to do this. A masonry chimney, especially if it is stone like the fireplace weighs many, many tons; the potential for huge damage to your house, not to mention its occupants, is almost guaranteed.
You might also consider that a fireplace adds coniderably to the resale value of your home.