I have old while pillowcases with tatting. I would like to color them to an antique tea look without using tea, since it could be harmful for longevity. I believe they are all cotton.

I can’t tell you exactly what to do since I don’t know what you have in mind. So I suggest that you get some bleached cotton muslin to experiment with first. Look at the reactive dye colors at www.dharmatrading.com. You could also start with Rit dye, which is easy to use and doesn’t require any additional chemicals.
have fun,
Jennifer