I was trying to respond very early this morning, but after typing in a fairly large comment, the entire thing disappeared and I decided it could wait until I was more awake. I have read a lot of different articles and "studies" which have been done over the last 30-40 years. Most of the studies have really boiled down to opinion pieces in spite of their title. This is a question that has a lot people supporting both sides, and don't think there is a clear answer to it. Most of the research indicates that we (can't find anything specifically about chameleons) get most of nutrition, vitamins, and minerals, from the foods we eat. I think most of us here would agree with that, which is why we make sure to gutload our feeders. Therefore, I don't think we really need to worry about nutritional deficiencies because out water has had it all removed (distilled). As far as the leaching of our minerals and so forth from our system, I find this highly unlikely, as most of what we need is intercellular (within the bodies cells), and our kidneys are remarkably good at filtering toxins and leaving fluid behind, which is why when we are dehydrated, our urine gets very dark, it is concentrating the things our body wants to get rid of into a small amount of water, to conserve the water our body still needs. I can't see how whether what we put in is distilled water or just a lot of tap water, is going to make a huge difference. Our military people out at sea drink almost exclusively distilled water during long deployments, with no significant health issues.
Ultimately, as far as chameleon health goes, I don't think it makes much of a difference. Where it can make a difference is in the maintenance of our equipment. Tap water can leave a lot of sediment behind in misting systems, which can mess them up, especially nozzles, as pointed out by a prior poster.
I live in the country and have well water that has some arsenic in it. Because of this, we have an RO system, which removes a lot of impurities (not distilled, but more than just a basic filter). This is what we drink and what I use for all of my animals. I have had some very long lived pets, that have stayed healthy their entire lives, so feel very comfortable with it. Curious to see if anyone has had a different experience or has found any recent research that indicates problems with distilled water!