Will the Earth's core cool?
Yes. It's kept warm by slowly  decaying primordial radioactive elements. They are: Uranium (238) and Thorium (232). The half life of the first is 4.4 billion years, the second around 14 billion years.

They heat the surface even to this date, and it's around 0.2K warmer than if they were absent.

Owing to their continued decay, this temperature differential will decrease .



Extension: On the most popular models (and David Hammen's comment :-) ), Uranium and Thorium are in the crust and   mantle, and not in the core. Despite that, their radioactive decay heats the Earth.