Most paints use pigments which will block light, effectively turning the lamp shade into a light blocking device. Better to buy a stain which is mostly transparent, but tinted to a darker shade that will still let SOME light through. Why not start new and just buy some cheap lampshades at Walmart or Target of some other store that would carry lampshades? Staining to cover up the dirt is not going to look pretty, and pigmented paint will block the light, so why not replace with new? Visit a Sherwin-Williams are any arts and crafts place and ask their advice.

Yes, I have done it before and it does not harm the lamp.