Some 42 million megawatts of energy reach the surface continually and are radiated into space as the earth cools from its initial molten state more than 4 billion years ago. No feasible amount of geothermal development could make even a small dent in this process. Furthermore, the earth’s heat budget is continually replenished by the radioactive decay of naturally occurring elements, and almost all of the energy associated with each decay event is converted to heat. Plus, the heat content of the geothermal reservoir rocks is continually replenished by conduction of heat from the earth’s deeper interior.
So a single nuclear power plant would be roughly equivalent to 0.002% of earths geothermal energy radiation. It doesn't sound like much, but for something as crucial as the earth's magnetic field, I wouldn't want to reduce it one bit.
I guess if the heat is radiated into space anyways and we simply capture that it doesn't speed up the core's energy loss, but if we start digging into the core and allowing energy to escape faster, we would effectively speed it up, right?
The radius of the Earth is 4000 miles. A 10-mile hole is thus 1/400 of the way to the center. You may as well worry that scratching the skin of an apple might damage the seeds.
Look at a graph of the temperature of the Earth as a function of depth. The temperature jumps tremendously below the crust, and is much more stable per depth below. The crust is the insulation of the Earth - and damaging that crust might affect the mantle is ways we cannot yet imagine.
The answer remains "we don't know".
The argument "it's big, we can't affect it" has been proven wrong in the case of the atmosphere, and again in the case of the ocean. It is no longer a sound argument.
Eruptions in Siberia, and later in India, not long ago geologically, released more heat in a (geologically) short time than we could use up in a million years, with no effect on the Earth's magnetic field.
I don't think we're even digging through the crust - there'd still be the mantle (~100x thicker) before we got to the core.
In numbers, I believe the deepest borehole is ~12km deep... and the Earth has a radius of >6000km. We're barely scratching the surface.
Since the layers are directly connected, it doesn't sound like to me that you'd have to dig into the very core for that to happen.
Of course at the current state of affairs the effects are miniscule, but it doesn't sound like a very good investment as something to potentially scale up in the future to where it will actually matter.
If it turns out to be a problem, how hard would it be to insert fuel back into the earth's core to sustain the magnetic field?
In another century, if we don't blast ourselves back to the stone age first (increasingly likely), energy from hydrogen-boron fusion will dominate wherever solar is impractical, and the geothermal wells will become too expensive to continue operating. So, either way, the whole process is an imperceptible blip.
Blasting our way back to the stone age will succeed in chopping off CO2 output suddenly, though. The climate could then return to normal in only a century or three, if the sudden change did not instead trigger an ice age or something.
You should worry instead about the effects of excreted pharmaceuticals and neo-nicotinoid pesticides on wildlife, excess fertilizer runoff on coastal ecosystems, and ocean acidification from CO2 dissolving to produce carbonic acid, making sea life unable to fix calcium for shells, destroying the underpinnings of the food chain and the whole ocean ecosystem. Oh, and global climate disruption.
We have no need to borrow imaginary trouble, we are making plenty of actually real trouble already.