It's not really possible to use enough geothermal for it to have a noticeable effect on the lifespan of the magnetic field. Even completely removing the solid crust of the earth would not really register.
So we could take out some percentage of the heat of the crust, thus lowing the total temperature by some amount, but then that loss would be replenished from further down in the core.
It looks like from the calculations elsewhere in the thread that it probably wouldn't even be physically possible to extract that much energy from the crust. We'd just run out since there would be some lag time in replenishing the heat from the core through to the areas of the crust.
But also let's hope it's not 0.001% because I'm pretty sure the core will cool more than that on its own over a shorter timespan than 19 million years.
19 million years is 1.9% of the billion year rate. So it suggests the core should lose 1.9C in 19 million years. That's about 0.035% of the total temp, using Wikipedia's 5700C as the temp estimate.
That's a great initial guess, given how much uncertainty is (i've recently heard core temp could be 4x the wiki number, and some people estimate the earth's core only has lost a few degrees since it formed, rather than nearly 500 degrees). Neat stuff, thanks for providing the opportunity to look in to this.
It's obvious that the answer is less than 100%, since a completely solid core would provide no magnetic field. Is there some way to tell whether it's closer to 0.1%, 1% or 10%?