Well CPU's are made of transistors. So one could ask if it is possible to make a transistor out of a superconductor.

The answer is yes. It is possible to quench a superconductor with an external magnetic field for example and make it an insulator. These things actually do exist but i forgot the name.

But switching that field still requires energy. So while it is possible to build such a CPU there would be still energy required to do any calculations.

Today's CPU's don't heat up because of internal resistance (mostly) but because there is current flowing due to the field effect transistors needing their gate voltages being raised and lowered.

Lowering that voltage means the energy must be vented somewhere which results in heat. That would not change if any part is superconducting.