It shouldn't hurt to cut peonies this early in Zone 5. Peonies stop growing after flowering and all the rest of the summer just store food in the roots. Right now, your peonies are putting out feeder roots from the tubers to feed next years growth, but the tops are just waiting to brown up. In fact, if it makes you feel better, I'm going to mow mine off this weekend...by choice.

Here is a link that might be useful: Garden Musings

I will also cut the mildewed peonies now July 16th but should I also put a fungicide or other on the plant after cutting the leaves?

Deborah you want to kill the spores by spraying even if you have removed the affected foliage and cleaned it off of the soil. What you do to eliminate it from coming back next year will help. Here in out mild winters it seems to live forever. Al