Is it safe to paint a lampshade?

Yes, there are fabric paints out there that work wonderfully for thing like this. Follow the directions and let it dry completely. https://www.amazon.com/Simply-Spray-Upholstery-Fabric-Paint/dp/B0077CB7MY

If your worried just use an LED bulb as they put out little to no heat. My Habitat for Humanity Restore had a huge stack of 60 watt equivalent LED bulbs for $1 each! I prefer brighter in my living room lamps but they work great in bedrooms, etc.

Hello, I recently spray-painted a paper fiber type lampshade from a thrift store successfully. In my particular instance instance with this lampshade I sprayed with the taupe gray on the outside and white on the inside is for greater light reflection and it made a nice contrast. I favor and have sucess with the spray paint formulations that have both paint and primer, in particular the Rustoleum brand.

Certainly a paper fiber or metal shade will accept spray paint better than a fabric shade.

Rustoleum recommendations

PREPARE:

Always work in a well-ventilated area. Cover the floor and nearby items with a drop cloth. Lightly scuff any glossy surfaces and wipe clean with a lint-free cloth. Use Painter’s tape to hide any surfaces you don’t want to paint.

PAINT:

Using slow, back-and-forth motions, spray paint each item with several light coats of spray paint, making sure to overlap each stroke slightly. Apply 2-3 coats of spray paint, waiting a few minutes in between each coat.

FINISH:

Let spray paint dry for approximately 1 hour before handling.

And you don't worry about the paint (even dry) being flammable being so close to a "heat" (albeit often small) source?