Interesting discussion. Here is my synopsis of the comments so far.
1. Minerals are not an issue when using tap water vapor to condition Bovedas or humidors since the minerals will not evaporate.
2. Minerals in tap water may have an adverse effect on humidification media due to the residual minerals left behind
3. If your tap water is chlorinated then the out gassing may be an issue for your cigars.
4. Distilled water is so cheap just use it and don't worry about any potential issues.