How do I repair spots of damaged hardwood floor finish?
Repairing these worn spots should be fairly easy assuming the damage is only to the urethane finish and not into the wood itself. This should be as easy as buffing the affected area with a green dish style scrubby to remove loose finish and slightly level the surrounding surface.  If the scrubby is not quite aggressive enough, use some very fine 220 grit sand paper gently by hand or with a DA or vibrating style sander. Without seeing the area and damage, it is hard to say how far you need to sand, but most likely do the area where the chairs have worn the finish to slightly into the non affected area. Be absolutely sure to get the same urethane product and finish (satin, semi gloss, or gloss) to do the overcoat. After cleaning all dust etc after sanding, apply a VERY THIN coat  of the new finish. Allow this to dry completely, buff sand the area again removing any dust that dried into wet finish, assuring it is extremely smooth to the touch. Repeat this process with at least 3 coats.  I recommend using a good grade foam brush or very good Purdy flared bristle brush. If the area is larger, consider a mohair roller.  A mohair roller is simply a very low nap roller avail at a good paint specialty store. You may see a slight difference in the new finish to the old surrounding area, simply because of the age of the rest of the room, however that is a small compromise compared to having the damage migrate into the wood which will require a much more extensive repair later.  Good luck