There should be no dust on the conductive surface of the CPU. If there is the heatsink/fan was not installed correctly. with that bing said there should be a layer of thermal paste or a thermal pad.

Plug the power connector for the cooling unit into the fan header on the board. This article explains how to install a heatsink on a CPU with the pin-grid array processor design used by most ...

One of the most common causes of a CPU fan failure is the accumulation of dust and dirt on the fan blades, the heatsink ... you should replace the fan as soon as possible. Cracks, dents or ...