Do I just leave it? PC case had rubber legs but it was stuck to the floor and when I tried to flip it, part of the wooden floor came off with it.
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What came off is the finish (urethane, polyurethane, or varnish), not the wood.
If left alone, it will absorb dirt, moisture and darken.
I would try a wipe on polyurethane and apply it with a q-tip. Your finish looks like gloss, so I would try that level of sheen. You can always dull it a bit with fine steel wool.
You should try to remove the rubber stain first. That can be done with a cleaner, such as Simple Green or Fantastik type cleaners.