In a recent video, Taras Kulakov the Crazy Russian Hacker demonstrates how coins react to being inserted into a block of dry ice. Kulakov is delighted by the way the coins shake, but does not offer an explanation. Instead, he asks viewer to explain it to him in the comments of the video.
The cause of the motion is that when the coin comes into contact with the dry ice, it heats the ice above its freezing point, and the frozen carbon dioxide sublimates back to a gas. That collection of gas pushes on the coin as it escapes, causing the coin to move and make different points of contact with the ice.