The content of mineralic ions in tap water is usually rather small compared to the amount that you take in with your daily food. Especially Calcium content varies wildly between sources, and nobody recommends calcium supplements in cities with soft, low calcium content water.
The common DI water you have from a tap in a laboratory building or buy in the supermarket is not safe for drinking, firstly and mostly because it is not tested. But who would want to drink it anyway? It is more expensive, often has a taste that varies between lame and slighly foul, and you do usually have a regular drinking water tap to use instead.
The only way you could run into problems is when you have little food, drink a lot (hot environment), and have nothing to drink but distilled/rain water. On the other hand even well water likely has not enough sodium to balance what you loose through sweating.
To answer your question: A big problem with distilled water is its corrosiveness to common household tubing. Regular tap water coats everything in a layer of calcium carbonate, which makes even lead pipes relatively safe. In a standard copper pipe, DI water will first slowly dissolve the oxide layer, and then a lot of electrolytic corrosion will take place.
So, to use distilled water as tap water, you either have to make everything out of PE (doesn't really work if you want hot water), or add calcium carbonate. I believe this is what's done in areas where desalinated water is fed into the communal water supply. You just have to mix it with a small amount of local well water.