Leaves curl on peonies under certain conditions. When they curl it is a signal that something is wrong with the plant. If this happens, take steps to correct the situation before it harms the peonies any further. If left unattended, the plant could eventually die.
Bring in potted peonies when weather conditions suggest temperatures might drop below freezing. This will prevent them from becoming frostbitten. Frost will make the plants die back for the winter. One indication of this happening is the leaves wilting and curling. For plants in the ground, you can place a plastic bag over them to insulate them from the cold. But you don't have to cover the plant--frost won't kill the a peony plant, and in fact "dying back" for the winter is perfectly normal. But if you aren't ready for it to happen yet, placing a plastic bag over the plant or bringing it in the house overnight will prevent it.
Water the peonies during dry periods. Lack of water can make their leaves curl up to avoid having to nourish them. If the plant gets too dry, the leaves will eventually fall off. This doesn't mean the plant will die, but it doesn't help the plant look pretty.
Check to see that the roots aren't damaged if the peony plant is getting plenty of water and there is no sign of insect damage on the leaves and flowers. Sometimes an insect or small animals might have disturbed the roots, preventing water from getting to the plant. A lack of water will make the leaves curl.
Remove any plants with mosaic, or "leaf curl." This is just one of the diseases that can affect peonies. Symptoms include curled leaves, patterns of yellow spots on the leaves, and shortened growth. Plants cannot come back from such a virus, and the virus will spread to other plants if not removed.