How much of an effect depends upon which part is changed and how large the change is. Pivot points at the ends of the linkage allow necessary suspension motion. Leaf springs function as both springs and linkage. Linkage affects suspension geometry and travel. Suspension linkage must have pivot points to work properly and bushings are often used as at least one pivot point.
Vehicle manufacturers use rubber because it works very well as a noise and vibration isolator.
If your vehicle has fairly high mileage on the original bushings, you may want to consider having new bushings installed. Some companies sell urethane plastic bushings to replace the factory rubber ones. Depending upon the location and hardness of the bushings, urethane bushings can cause a harsh ride and road noise in the vehicle. Depending upon where the urethane bushings are placed they can also cause handling problems in some suspensions. A suspension does not have to have anti-roll bars, however they are quite common on many vehicles.