Copper mugs are famously used to serve a long-running popular drink known as the "Moscow Mule." However, using an unlined copper mug is typically not advised. Most copper mugs used for food and drink consumption have tin or stainless steel lining. Without lining, copper dissolves into food or drink at extreme temperatures.
Though it doesn't address mugs specifically, the Clemson University Cooperative Extension website notes that the Food and Drug Administration advises against using unlined copper in cooking because of the risks of dissolution. Side effects indicated include nausea, diarrhea and vomiting. While the information doesn't specifically address consuming cold beverages from copper mugs, it is safest to only copper mugs that have a lining or a disposable or washable insert.