Hazel - I just did some Googling. I already knew that most invertebrates (like my shrimp) are very sensitive to copper. Sadly, it sounds like fish are also more sensitive to copper than people, so if you are already having to filter your water for you, it is likely unsafe for fish.
If you decided to run a small tank with distilled water, be aware that some minerals are necessary and you would have to put back some of what you take out. That's the extent of my knowledge of what distilled water tanks require.