The short answer is, it depends on the type of buttercream frosting you're making. If you're using a high-ratio of sugar to butter, it's likely to be more stable at room temperature. But, if you're using a low-ratio, it's probably a good idea to refrigerate it to prevent bacterial growth.
Think of it like leaving a pizza out at room temperature - it's not the end of the world, but it's not the best idea either. Bacteria can start to grow, and before you know it, your frosting is ruined. So, it's always better to err on the side of caution and keep it chilled.
