Most often, with older oil fields that relied on simply pumping oil out of the ground as long as possible – nothing at all. Sometimes the ground subsides, whether that’s even measurable or not (it probably would be with good enough instrumentation, but there’s been no call to do so, I think).
Or in more recently developed fields, or fields that are being “recovered” after having been abandoned sometimes decades ago, what @thorninmud and @majorrich said.
Sometimes the assumption that “nothing happens” can be correct, if the oil (or gas) is pumped from a stable enough strata, but I don’t think that is common.
This doesn’t account for the times that we fill voids with the rusting hulks of used pickup trucks, of course.
A related question that you may not have considered is “What happens when water is pumped out of huge aquifers?” The same thing happens as described by @thorninmud – the rock just gets drier, but there is so much water that in many cases the ground does subside, and it is of interest and measured. I don’t have any current links, but I’m sure that if you look for “Oglalla Aquifer subsidence” you’d find plenty on your own.