First things first: Human activity is not tapping into the heat of the Earth's core. At best, we're tapping into the heat differential between the surface and tens of meters to perhaps a few kilometers below the surface. Temperature in general increases with increasing depth. We humans don't have the technology to penetrate more than a few kilometers below the surface of the Earth, let alone the technology needed to penetrate the six thousand plus kilometers needed to reach the center of the Earth.
That said, the Earth's core does produce heat. It retains a bleep ton heat (read a crude four letter word instead of "bleep") from its initial formation. This initial heat came in two forms. One was a result of collisions. Even more heat was generated when the Earth separated into a core, mantle, and crust. This is where the bleep ton comes into play. The Earth has only had 4.5 billion years to radiate away that huge amount of heat. That's too short of a period of time for that huge amount of heat.
Regarding heat production, the Earth's core produces heat via the conversion of molten material in the Earth's molten outer core to solid material in the Earth's solid inner core. The Earth's core may also produce heat via radioactive decay of material within the Earth's core, but this is highly debatable. The four main long-lived radioactive isotopes (uranium 238 and 235, thorium 232, and potassium 40) are chemically incompatible with migration to the Earth's core. are chemically incompatible with migration to the Earth's core. The former source of heat (an increase in the size of the Earth's inner core) is widely accepted. The latter (radioactive decay in the Earth's core) is anything but widely accepted.