Hi Wes, a mistake made by many people is they will get and old brass oil lamp or candle holder etc that has a lovely aged patina and they polish back to a high gloss finish and effectively reduce its worth.
I would not add any polishes or wax to your petrified tree I would probably just remove any loose dirt or dust with a soft bristled brush and maybe give it a wipe over with a lint free slightly moist cloth. Then wipe it over with a dry, lint free cloth.