Hi Stephanie, the how to is going to depend very much on the fabric that the lampshade has been covered in, but perhaps this article will help - https://homeguides.sfgate.com/remove-water-mark-lampshade-59420.html
Here's an article that might help - https://homeguides.sfgate.com/remove-water-mark-lampshade-59420.html
What is the fabric? If silk, do not subject to any moisture.
If it's silk, expensive, or antique, see if your dry cleaner can do it!
Baking sodas and peroxide if it is normal lamp shade material. Use s fine mist sprayer worth peroxide, spray til damp sprinkle with baking soda then lightly brush with nail brush. Let dry and vacuum with small hose attachment or if delicate you can brush off with a duster.