What's the shade made of?

You might want to consider how changing the white shade will alter the color of the light from the fixture first.

Tea or coffee stain may work to make it not such a bright white. But you may not like result then what? I'm assuming it's fabric. Although not all fabric takes stains. I once tried to dye some medium blue fabric black for a Halloween costume. Would not take dye at all. Was probably polyester kind of fabric.

Maybe you can line the inside of shade with a sheer fabric that had some color to it, might be a interesting effect.