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Hmmm, might you not find that the spinning wheel causes a resistance to the car pitching (ie forward/back, left/right)? I'm not sure of gyroscopes, never have looked at them properly, but I think this might happen. What effect it would have on the car I really don't know- not sure if the body of the car ever has to pitch/roll large angles quickly.
It might slow down transfer of weight between the tyres as the car went from turning left to right (or vice versa) or from accel to brakes. I doubt this can be a good thing, but then again surely the effect can't be a large one.