I've been drinking distilled water for years. It does not show any signs of having reduced any mineral content from blood test results. At worst, it might not be as good for your teeth because there is no fluoride in it but if you brush daily with fluoride toothpaste or use mouthwash with fluoride then it's not a big deal.
I can't think of any specific mineral you would miss that you can't take a multi-vitamin to get. In places where drinking water isn't so pure or is heavily polluted, distilled water might not be so bad.