On my unpainted tungsten weights, I have some build up or corrosion on them >--hard to explain.  It does come off with a little scrubbing, but there are stains where the build up was.

I've clean them and my storage box, but it returns after time.  Any ideas what this might be or how to clean them so they stay clean??

I'm guessing tarnish, sort of like on silver. From odoing a quick search on google, Tungsten is made in two fashions, cheaply with Cobalt which can tarnish and leave brown stains, or Tungsten with nickel which is more for jewelry and such.