I worked in a nursing home that believed that no matter what a floor doesn't need to be stripped more than once every 3 years! I've ran into that problem many times over.
What I do is scrub the room and in the affected area only, spray a light mist of baseboard finish remover aerosol. Don't let it sit too long. 30 sec to 1 min max. Run the buffer over it and rinse. Repeat as necessary. It'll remove just enough finish to get rid of the contamination, but won't remove all of the finish so it'll blend in. Then apply your new finish. This also works for deep scratches and stains.
If it's made it's way all the way to the tile and has began wear the tile itself down, what I do is apply an extra layer on that area in between the layers I put on the whole floor. That way it will have double the amount of finish on it thus building that area up higher than the rest of the floor and blending it in. This usually works if the damage to the tile isn't too bad.
If you have a stain on the bare tile that won't come up with scrubbing, put a little ajax or comet on it and scrub. Just be careful not to scrub too much and damage the tile. Rinse the area and continue to scrub the rest of the room.
I hope this works for you. Let me know how it turns out.