Let's start with the simple stuff. Peonies are perennials and yes, once it looks dead cut it down tot he ground.
- Some horticulturalists argue this should be done in the spring to protect the roots from cold.
- Others say removing all dead material in the late fall reduces the chance of disease being carried through to the next season.
That brings us to the hard part.
There are a number of fungal diseases that can cause spotting on peonies; the worst of these is Phytophthora Blight. If this is what you have, i should be cut out of the plant and removed. Then move the healthy stems to another area since the soil is infected.
Other fungal diseases may be controlled with fungicides and/or may need to be treated similarly to P Blight above. Yes, these fungii will interfere with bloom time and possible re-blooming.
For more information:
Read "Diseases of Peony" a post on the University of Minnesota website.