I am doing a school project and need to know...
Will the Earth's mantle or core ever cool? If so, how?
Yes. It is kept heat by slowly (very slowly) decaying radioactive elements. They are: Uranium (238) and Thorium (232). The half life of the first is 4.4billion years, the second around 14billion years.
They heat the surface even now, it is around 0.2K warmer.
As they decay, this temperature will be lower amd lower.
Extension: On the most populer models (and David Hammen's comment :-) ), the uranium and the thorium is in the crust and in the mantle, and not in the core. Despite that, their radioactive decay heats the Earth.