How do you get wood stains out of wood? Or, how do you get water or hot plate stains out of the wood?
Water stains can be removed from the bare wood with oxalic acid. Mix a saturated solution and apply, and wait a few hours to overnight. Wash off well with clear water, and allow the wood a few days to dry out, then touch up the finish. If this fails, you will have to sand it out.
As for the burn you will have to sand it out, with 80 grit, then 100 grit to smooth, then touch up the finish. And I hope you know what your finish is. If not read my article available in the search box at the top of this web page about floor maintenance (the second part).
Oh, and if you find that the stains are too deep for either method you may have to repair the floor. Read about this also available in the search box at the top of this web page.