Somebody correct me if I'm wrong, but I suspect the most likely problem with some tap water and any kind of mister would be mineral build-up that would clog the works, even if there wasn't anything in the water that would hurt the animals. Tap water varies enormously; in some areas, shower heads get choked off with crud in a year even though the water is perfectly fine to drink, and in other areas you never have to worry about that.

Whether your tap water is safe for herps (or people) and whether it will damage misters (and pumps, hoses, showers, teakettles, etc) are two different things.

You don't live in Flint, MI, do you?