In most cases it is OK. Lots of traditional foods are made in copper. I've eaten food made in raw copper my whole life and have yet to get ill. This includes apple butter, which is acidic and cooked in copper for hours and even gets certain levels of green verdigris in from the long exposure/reaction time.
As for cheese, in commercial applications I have found that the standard is largely, copper can be in contact with milk products as long as the pH is 6 or above. If your pH in the vat drops into the 5's as in cheddar and such, you are not allowed raw copper (tinned copper would likely be allowed). But this comes with the added requirement, they still only will let you do this if you are making some kind of traditional cheese where copper is important. This basically means, most states will let you use copper for Alpines, and that's about it. This is probably because, being Swiss, we all fussed about having to do something other than the traditional way and got an exception as a result.
For just about any other use, I don't see any real advantage to copper. But remember, this is a guy whose contact surfaces are mostly sealed wood.
The materials used, outside of the vat itself, are not that critical. What's more critical is the care you give them. I don't like stainless steel because it lulls people into a sense of false security -that is, I see people not working as clean, because 'I have stainless steel' this is not universal, but it does happen.
I also don't like it because it is cold, and soulless. I like to be surrounded by wood. If I am going to be spending my days in this room, I think it should be pleasant.