No. There's no work being done from the rotation itself, if the object is perfectly rigid. Of course, in the ideal case there is work being done due to tiny stretches all over the place, but that should all cancel out because it both stretches and contracts to keep the same shape. With no overall radial motion coaxial with the force, there is no work being done.

Also, a spinning sphere, or a cylinder spinning around its own axis, will not emit gravitational waves. But something like a rotating cog would emit gravitational waves.