You've hit upon a fundamental problem that continues to dog rotary-wing aircraft; Honing & Lagging: Say you have a single-rotor helicopter with standard clockwise turn; The left-side (the honing side) will always be faster than the fuselage of the helecopter while the right-side (the lagging side) will always be slower! As level speed increases, this honing and lagging also increase so the left side blades are going fastest of all, the righ-side blades slowest of all and the helicopter itself is somewhere in the middle! The left-side blades will reach supersonic blade tip speed when the helicopter reaches approximately 350mph! But with the speed of the main rotor blades the left-side will reach supersonic speeds actually before that horizontal straight-line speed is reached. Should this happen the helecopter would most likely be totally unflyable; At transonic speeds the honing blades would be violently flexing as the shockwave attached itself to the top of the blades. As soon at the blades came around to the lagging side they would go subsonic but due to their lagging behind and the forward speed, they would make no practical lift at all. Assuming some way could be found to prevent the total break-up of the blades, about the only way the helicopter could stay in the air would be in a hover situation and even then the horrific noise made by those supersonic rotor blades would be deafening for miles around.

The simple, and not so simple answer is yes. It can exceed the speed of sound if spun fast enough. However, there are several issues. Because the blade is rotating, and not flying straight into the air, then the outer tip will be moving through the air faster than the base. In fact, the airspeed of the blade will increase as you move out. So what will happen is that the tips of the blades will be the first to reach mach 1. A shock wave will form at the tip of the blade. As the blade increases rotational speed, the shockwave will move along the blade as more of the blade goes supersonic. The big problem with this is that the blades really aren't designed to withstand the stresses of supersonic travel. They will end up disintegrating. It also means that a portion of the blade will be 'transonic' (at or near the speed of sound). In this region there are problems with airflow and controllability, which will severely hamper the performance of the blade. Loss of lift and poor control will be major symptoms. Noise is the other issue.