does sunscreen hurt spray tans?

No, your sunscreen acts just as if any lotion would. It moisturizes the skin, and will not take any of the tan off.

Boshena's comment: Actually, SPF30+ or higher can have some effect on spray tan. I think that it is bacause some sunscreens contain artificial oils and those can take spray tan off when highly concentrated (just like baby oil can).
So, use lower SPF+ sunscreens and stay out of the sun when you have a spray tan on.