I have read that one should not plant a new peony in the place where you have removed another one. Is this so, and how long must one wait to use that location?
Can You Plant a New Peony in the Same Place as an Older One?
If a peony (or any kind of plant for that matter) has been removed because it was diseased, you don't want to plant another one in the same spot, to protect the new one from similar infection. You should not plant any plant in that spot that is susceptible to that particular pathogen. In addition, gardeners often rotate crops to minimize soil exhaustion. Similarly, a new species of flowering plant might do better in that spot. Of course, good soil amendment habits pretty much take care of mineral depletion: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/improving-garden-soil.htm
The time required to rid the soil of a particular pathogen varies with the pathogen and treatments. You should be able to get more information on that from the Extension Service. https://www.gardeningknowhow.com/garden-how-to/info/what-is-extension-service.htm