BaIrO3 is a novel insulator with coexistent weak ferromagnetism, charge and spin density wave. Dilute RE doping for Ba induces a metallic state, whereas application of modest pressure readily restores an insulating state characterized by a three-order-of-magnitude increase of resistivity. Since pressure generally increases orbital overlap and broadens energy bands, a pressure-induced insulating state is not commonplace. The profoundly dissimilar responses of the ground state to light doping and low hydrostatic pressures signal an unusual, delicate interplay between structural and electronic degrees of freedom in BaIrO3.