The fiberglass filter “sock” that lays in the bottom of the tank and serves as the fuel pick-up for the pump will allow almost every last drop of fuel to be drawn from the tank…At that point, fuel pressure is lost, the engine dies, and the pump shuts down within a few seconds.
So back to the OP’s question…When you have 1/8 tank of fuel or even less, you are NOT damaging your fuel pump. These pumps, like everything else in the vehicle, have a “mean-time-before-failure” designed into them, the point at which 50% have failed and 50% still operate… That failure point is built in…Nothing lasts forever.