If I were moving a tree peony I would wait until fall to help it stay with its more natural cycle. If it is an emergency, in the spring you could do it before the growth really gets started. Dig wide and deep to lessen the root damage and replant at proper depth. Also, if it was grafted, on a well established plant with its own root system now, you can sever the grafted stock as it is no longer needed, or re-use that stock for further grafting.
Sometimes moving other things instead of the tree peony may actually be easier and give it room to show off again.
Sharing here is really great. I consider myself to have a love of peonies but not enough confidence and expertise to always shout out. Have learned a great deal over the past few years from my local peony society and through that I have been given experiences to further my knowledge, which I thoroughly enjoy and am happy to pass along to you. The tree peony grafting is a great example. Will share my results if it ever warms up/dries up enough to get the initial growth going.
Thanks everybody for sharing and posting your own experiences here. I always love reading about them and seeing them.