Does removing oil from her stabilize or destabile the structural rigidity of planet earth
Neither. The amount of oil that could be removed isn’t significant in terms of the mass of the Earth or its crust.
On a purely local scale, pumping out oil can cause some settling and microquakes. In quake prone regions, like the LA basin, oil pumping has been blamed in making some quakes slightly more damaging. But we are talking about the very fringes of something we can measure and analyze.