If the shaft can not move an any direction except to rotate the magnet will have little effect. If the shaft were to rotate due to some other force, there would be a braking force due to eddy currents induced in the shaft. The resistance of the shaft would tend to minimize that effect. The effect would be maximized if the shaft would be copper plated.
Added re change of question:
Making the rotating object non-homogeneous doesn't change the answer much, whatever the internal details and position of the rod, the equilibrium net force will be to displace the rod rather than making it rotate. Only a continuously moving magnetic field can make an object move continuously. The magnetic field must move in a way that will not allow the object to reach an equilibrium position. I think you can analyze any system of magnets and ferromagnetic objects by diagraming the forces and resulting motions and determining that the object will only move to an equilibrium position. You know the outcome because of the conservation of energy principle, but you can probably demonstrated it also.
Added re comment
Switching magnets on and off in a way that avoids getting stuck in equilibrium is essentially making a switched reluctance motor. That will work.