Well, I know that people have been very sick when using copper coffee pots that have had their tin coating deficient. Humans need a small amount of copper but apparently some copper vessels have more than that dissolving.
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.
It is apparently partly a question of pH of water used, high is good. There can be a passivated layer formed on the inside but that is not safe. So if no lining better not use it, seems to be the message on several sites.
If I remember correctly there are some real chemists on board.
You don't say if it is a cooking kettle or a tea kettle. If it is a cooking kettle it is okay to use providing you do not cook any acidic foods that will leach the copper. A tea kettle is fine for just boiling water bigbear. I use a brass cooking kettle for food & water, but it is tin lined.