Although I am sure you will want to refute it, no.
The tarp is dry when it is placed over the pile. Dampness does tend to cook out of all but the most bone dry piles. These are not season long static situations. There is a lot of turnover. Even if there wasn't, tarps are routinely used on baseball infields, plates, and mounds. While the tarps sometimes get a little musty, the infield dirt is fine. Granted, we are not trying to grow anything in infield dirt.
Bottom line is I have never had a problem using pre germinated seed with tarped sand-compost mix.