Never had a problem using black plastic, it actually speeds the process by drawing in heat. Anything you use should be allowed to air out. Remember to let the dirt cool before storage or you will get clumping. Many tarps are woven material and allow heat to escape lengthening the process.
Funny thing when people say plastic is made from petroleum and will contaminate the dirt, so are the totes and buckets most people store their dirt in as well as the paraffin wax used in the freeze proofing process.
I've said that as well and recommended using pails that where used to feed livestock and such. But then not all animals are afraid of human scent. I've have them shy away for having bad wax dirt but then made sets bare handed and caught them. I'm not one to go overboard on trying to be scent free but then I do believe one shouldn't be careless either.