When deadheading peonies, is it best to treat the spent blooms like end-of-season foliage (i.e., discard them in in the trash), or is it OK to leave them under the plant?

Generally speaking, it is always best to discard any material trimmed from a plant. Area sanitation helps to prevent various types of diseases.

I have done both. I have not noticed any difference. Those damn peonies just come back year after year.

With the rain that we have had this year in PA, I have seen black spores on my peony buds. No botrytis yet but I would not chance it. I would just discard the spent blooms in the trash to prevent any fungal problems brewing in the soil or anywhere else. Some peonies are very susceptible to fungal problems esp. tree peonies.

Knock on wood, but I have never had any fungal problems on any of my tree peonies, ever. Herbaceous ones have been affected from time to time but never the woody peonies/ tree peonies.