No, earthquakes are caused by tectonic plates shifting, they do it all the time, nothing to do with mining or drilling. The earth's not going to cave in.
Earthquakes are caused by two continental plates get stuck on each other as they move, then being released with a sudden jolt. IMO they're so big and in comparison the weight of the oil removed is tiny so has little effect. But I think some companies do/ are looking into something where they capture greenhouse gases and put them back into the empty pockets to stop them being released into the atmosphere.
Oil in the ground is not in large open spaces like a cave filled with oil. It is in rocks like sandstone etc with the tiny spaces between the particles filled with oil, that is why is usually quite slow to extract from individual oil reservoirs and after it is extracted the rock is essentially still the same as before.