I live in Baltimore, MD, growth zone 7b. I grow peonies in 7 or 13 gallon black plastic nursery containers. The containers and their peonies remain out-of-doors year round. I have been growing in containers for about 11 years. I have several hundred varieties. I grow herbeceous in 7 gallon containers (>600 containers, >400 varieties). I grow Intersectionals in 13 gallon (>200 containers, >70 varieties), and tree peonies in 7 and 13 gallon containers (>150,>100varieties). I believe 20 gallon containers would be best for the tree peonies but they are too heavy for me to handle. The larger the container, the easier it is to maintain moisture level, it reduces the frequency of dividing and repotting, and provides more room for the roots. Larger containers also do not get as hot in the summer, and I suspect may work better in very cold winters. I have guessed for years that containers in hot locations would allow the roots to get more chill hours in the winter (just a guess). If someone lives in a very cold place, and they wished to grow tree peonies they may be able to move them to a garage in the winter for protection (probably still cold enough to ensure dormacy).
I have grown Sarah Bernhardt, Fiesta Maxima, Karl Rosenfield, and an unknown single pink in 7 gallon containers for 11 years. I started with a few and now after dividing and repotting every 4 years I have about 150 pots of those varieties. They really need repotting again but I have no use for the 900 pots that likely would result. So I have put off repotting until 2010.
I started growing Intersectionals about 5 years ago. I found they did not grow well in 7 gallon containers. I started growing Intersectional in 13 gallon containers about 3 years ago and they did very well. So last year I put all my Intersectionals in 13 gallon containers.
I started growing tree peonies 3 years ago. I grow some in 7 gallon containers. But due to the expected growth I want to switch them all to 13 gallon containers next year.
I use good slow release fertizer most of the time. I have used rose fertilizer (due to a good sale). I never had any bloom problems even though it was high in Nitrogen. I do have to spray for fungus since I have so many plants close together. The most critical item is maintaining the appropriate level of moisture. I custom mix all my soil. My mix depends on the quality of the top soil I start with. I go to every garden store until I find some who has cheap 40 pound bags that have some clay, sand, and a good amount of organic material, I then add 40 pounds of bagged humus/composed cow manure to 160 pounds of topsoil. The right topsoil is hard to find here in Baltimore, so when I find it, I usually buy several hundred bags. I mix in 4 oz of bone meal with each 200 pounds of mixture. The resulting soil must balance between reasonable drainage and water retention.
Plant divisions normally and water in.
Fertilize a couple times a year. Nutrients will wash out. You have to water pots about 2 times each week in the hot weather. Watering is critical. Peonies in containers will dry out very quickly if it is hot and or windy. However, I try to leave my containers in place to allow them to grow out the bottom into the soil. The plants then get access to the nutrients that washed out and have access to moisture in the soil. I plan to put in a drip system. A drip system would apply the water slowly and thus be more effective, waste less water, and same me about 9 hours of work a week.

It is important to note that peonies with lots of leaves will shed rain water to around the pots. Thus you still have to water them. If you have a long slow rain of 1 or more days the plants will get much of that water.
A major problem is if the containers are near trees or shrubs. The term "near" is relative. Trees send roots into the containers from the bottom. If a container is too close to certain trees or shrubs they will send invading roots into the containers from the bottom. I have found 7 gallon containers near a maple that were over 50% tree roots within 1 year. The plant died as a result. I now have to move those pots that are near trees. I only move them enough to break the invading roots and move them a few inches. I have to do this 2 times a year. From my experience locust trees are the worst (40-50 feet and I still get invading roots).
I find that growing in containers is very different than growing directly in the soil. I can move the plants any time I wish (although I don't). It fairly easy to dump the container, divide and re-pot. The cost of the containers and soil does add up. But most people would not wish to grow as many as I do.
I also grow 100's of other plants in containers, including day lillies, astibles, hosta, iris, hibiscus, roses, and butterfly bushes. I have found that hellebores and lillies do not grow well in containers. Day lillies, iris, hosta, and astibles grow very well in containers. Shrubs grow too well in containers.
I do not know how well peonies grow in containers would survive in very cold places. In hot places I would advise against black or dark containers.
Peonies grown in containers tend to lose their nutrients due to water flushing the nutrients out the bottom of the pot. I fertilize two times a year. I suggest slow release fertilizer with micronutrients designed for bulb growing. However I have never had a problem using slow release rose fertilizer for my spring fertilization. You need to use something with micronutrients at least once a year. I use what I have available for the first feeding. I am very careful if I use a regular fertilizer like 10-10-10, I apply it prior to sprouting time. My second feeding, is where I use a good coated slow release bulb fertilizer, that has micro nutrients. I follow directions. An additional concern, with the reduced surface area in a container, you have to be very careful not to get the fertiziler granules on the canes.

For those who may have wondered, my yard looks like a commercial nursery although I do not sell plants. I am just a very compulsive collector. Especially with Intersectional Peonies.