I love Moscow Mules and we registered for Copper Mugs so this scares me!
According to Iowa's Alcoholic Beverages Division, you're really not supposed to have copper come in contact with acidic food or drinks and Moscow mules are acidic.
This means that as you drink it you can absorb just a little of the copper in the mug and pass it along to yourself.
So if you drink them enough, you could wind up with copper poisoning and that can cause fatigue, fainting, and depression.
Source: Metro