Oh I hate this topic. None of us realllllly know the answer coz we have never tested for copper in our hot water AND we all have systems of a slightly different age so what is true for us may not be true for anyone else.
Id say, If you havent replaced any pipes recently, and your boiler is not new then any copper that is likely to be free'd up and dissolve into the water would have gone by now.
To play it safe if you're worried - or if you have shrimp in the tank (Which are very sensitive to copper) use a dechlorinator that also binds heavy metals. I think stress zyme does this although others do as well. Simply read on the packet. Also, only use enough hot water to take the edge off the cold. Fish are not bothered by a drop of a few degrees, simply pour the new water in by the water-output of the filter so any cold is "diluted", and take your time.