Wood and metal chair rungs wear down due to shoe scuffing. Both types can be protected using a multi-layered approach. Discourage table guests from from resting their shoes on the chair rungs.
Most chairs are sold with a thin finish coat over the wood stain or metal painted surface. The finish coat will only tolerate minimal use. This is why the arms and seat backs of chairs show wear before the back of the chair and legs. Usually the arms are gripped more, washed more and the finish coat deteriorates faster. The fastest and least expensive way to protect chair rungs is to lightly sand the rungs and spray them with several coats of clear paint finish. Allow each coat to dry completely. You can also touch up the chair arms and the top of the seat back at the same time.
For chair rungs that see a lot of abuse, a simple fix can prevent damage. Look for transparent flexible tubing in the diameter of your chair rung. Cut the tubing to length and then split it. Slide the tubing over the rung with the seam toward the floor. Use a zip-tie at each end to secure the tubing. The transparent tube will allow the color of the rung to remain visible.
An old-fashioned remedy to protect rungs involved wrapping the rungs in rope, about 1/8 inch in diameter. Use an upholstery tack to attach the end of the rope 1-inch from the end of the rung. Wrap over the rung from one side to the other. Keep your wrap tight. This will also add a nice textural element to your chairs.
Fabric can add a more formal and intentional appearance to a rung cover. Look for pipe insulation foam in the diameter of your rungs. These already have a split. Cut and fit the foam to the rung and zip-tie the foam in place. Sew a simple cover with hook-and-loop tape closures. Use a fabric you can launder when it gets dirty. These will keep your chairs looking fresh and clean.