There comes a time when age and the elements have taken a toll on brick and mortar. An unstable chimney, even if not used, can be a tremendous liability for the house and its occupants. Yes, it can fall down! However, it may be better to dismantle to a point and rebuild, thus restoring the original appearance to the house and eliminating a safety hazard.
You might need crown resurfacing or a cast concrete crown and not a total chimney rebuild.