I don think it does, the torque is applied to the outermost fibers of the shaft, which is why a lot of shafts aren't solid but hollow. It is also cheaper to make.
Like it has been mentioned, it does not effect the maximum torque allowable. What it does have an effect on is the stress distribution and the deformation in the shaft created by the torque because they are a function of the polar moment of inertia.