It's an interesting question. The boiling water will, undoubtedly, take up more copper than cold water but probably very little once any calcium forms on the surface. On the other hand, the cold water runs through tens of metres of copper piping and, on occasions, sits in the piping overnight or longer.
Coffee is acidic and may well have other chemical reactions with copper that could cause sickness.
Would I use it every day for every cuppa? probably not. Would I use it occasionally at a steam punk camp? yes, definitely.