Yes, at least in the short term and in the area immediately around (and at the depth) where heat is being extracted. Rocks are not great conductors, so in extracting heat from rocks in a geothermal power plant (usually through cycling a fluid to depth and then back up to the surface), the temperature of the rocks at the target depth in the vicinity of the plant will decrease through time basically because the process by which we extract heat is much more efficient than the process by which the rocks are reheated by conduction or movement of heated brines. This is one of the reasons individual geothermal power plants have lifespans (along with the progressive corrosion of, and precipitation of various solids in, the pipes and pumps involved in the geothermal plant, depending on the type of plant). The lifespan of a plant can be extended if less power is drawn from it (i.e., you can get a lot of power for a short time or less power for a longer time), but eventually the productivity of the plant will still decline through time (e.g., Budisulistyo et al., 2017). Similarly, there are mechanisms to try to replenish heat to slow the degradation of the resource, e.g., one proposal is coupling geothermal and solar heating where you heat fluids at the surface via solar energy and then cycle these hot fluids down to offset some of the heat extracted as part of generating geothermal power (e.g., Wendt & Mines, 2014).

In terms of long-term or large-spread geologic effects of this? Not really. Because rocks are poor conductors, the area of cooling will be relatively localized around the horizons being exploited by the plant and the area will be reheated, but on a timescale that is significantly longer than the typical few decade lifespan of an average geothermal power plant.

I think the other comment is probably more pertinent to what you were asking, but it's also interesting to note that geothermal heat comes from decaying radioactive isotopes and tidal heating. Both of these also reduce over time, as isotopes decay into more stable ones and orbits circularize, so these sources will also be "used up" too, but on geological time scales

Yes, but when that happens you can just drill deeper. But of course, there lies the crux of geothermal power and why it isn't really taking off. In most parts of the world, that drilling part is just too expensive to pay off. Would be really neat if we could drill cheaper though, it's a stable and near limitless energy source that is available everywhere. When the economics improve I'm sure it'll take off as an energy source, but when that might happen is anyone's guess.

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