Unsrung mass has the effect of transfering road-shock to the car. It has inertia and doesn't want to respond to changes in the road.
Also, it slows the tires downward return to the pavement causing a potential loss of contact with the road.
Consider a go-cart; All mass is unsrung and all road irregularities are tranfered directly to the car.
If it were possible to have 0% unsprung mass the tires would quickly respond to road changes and return downward without delay giving a virtually perfect ride.
Seeing as perfect isn't possible, minimizing unsprung mass is highly desirable.