No, you should not refrigerate Cinnabon rolls. Refrigeration can make them stale quickly. The starch molecules in bread recrystallize in cold temperatures, causing the rolls to lose their soft texture.
Instead, keep them at room temperature. Place the rolls in an airtight container. This way, they stay fresh for up to two days.
If you need to store them longer, freezing is a better option. Wrap the rolls tightly in plastic wrap and then in aluminum foil. You can freeze them for several weeks without losing quality.
To enjoy them later, simply thaw the rolls at room temperature. Warm them in the microwave or oven for that fresh-out-of-the-oven taste.