Of course it wouldn't - and to be honest here it's not even hard to imagine. If the lever was absolutely hard and could not bend (normally it must bend - the speed of waves in material is also limited) you'd need more energy to move it faster.
To imagine this, take your bicycle and try to rotate one of its wheels. As you put your finger closer to he centre, you need stronger force (but same energy of course) to get the wheel moving. So the distance from the centre ($r$) matters somehow obviously. And when you try to give the mass at the perimeter speed over $c$ (lightspeed), you are in the same situation as here.
However, what may be interesting for you, unreal objects may bypass the speed of light the way you tried to do it with lever. One example can be a laser beam, quickly swept over distant planet. The "laser dot" can move faster than light, but because it does not cary any information, this fact is useless (the light from laser still travels normally and takes its time to form the red dot.
Note to the deleted comment by somebody:
I did not take rigidity into account on purpose. I wanted to explain that regardless to the excellence of the material, desired effect would remain unachievable. Doing this, I actually assumed that the speed of information in the rod in above example would exceed lightspeed.
If we want a realistic example, we end up with a spiral instead of rod - but we'll be able to move it at any speed provided we have sufficient power (which will never need to be infinite).