Does radiation cause the earth to lose heat?

Yes, as the heavy radioactive elements (e.g. uranium and thorium) that have the longest decay chains are concentrated in the inner/solid and outer/liquid nickle iron cores most of the heating occurs there. But radioactive decay heating also occurs in the mantle (where it drives convection cells that power plate tectonics) and to a lesser extent the crust. Lighter radioactive elements (e.g. carbon-14) that decay directly to stable elements are concentrated in the atmosphere, while they do produce some heat it is totally overwhelmed by heat from the sun.