There is no reason why you couldn't build a motor using superconducting magnets, or build a simpler homopolar motor using a length of superconducting wire. There would be no heat dissipated due to electrical resistance, but of course there would still be mechanical resistance due to friction on the moving parts. The energy needed to overcome this, and to drive any load to which the motor is attached, would come from the power source, just as in any other electrical motor.
Thermodynamically, what must happen is that the current drops to zero as the motor accelerates, as the energy in the circulating current gets converted into kinetic energy. Presumably this is because the acceleration causes the charges to experience a force component opposite to their direction of travel through the wire. My electromagnetics is too rusty for me to be confident about the mechanism that causes this - but the key point is that what happens is that the force that stops the current is due to the fact that the charges in the wire are moving through a magnetic field, which is different from electrical resistance.