The overriding principle here is that everything tends toward equilibrium. I think that that has been agreed upon. However, in terms of what pure RO water has on life, is that pure RO water exposed to anything will not be in equilibrium. That is because, equilibrium allows some portion of solids to be dissolved in water -- this is all solids. Now, it may be an incredibly small amount dissolved, i.e. RO water in glass only has a few molecules of glass in the water at equilibrium. On the other hand, a pinch of salt or sugar dissolves very quickly in a cup of water. But, as most of you know, if you keep adding salt or sugar, eventually the water becomes saturated -- or, in the terminology I am trying to introduce, the salt or sugar solution is at equilibrium with the solid form.
What happens with RO water, when it is exposed to higher life forms' tissues, (i.e. fish or human tissue), is that the pure water is not in equilibrium with the minerals in the tissue. That is, the concentration of a mineral like calcium is not in equilibrium. So, the tissues of the body gives up calcium so that equilibrium is reached. I chose calcium for my example, as I think it is the most critical example of a mineral you or your fish do not want to lose. Everyone knows how important calcium is bones, but calcium is also critical for cell membranes. Without enough calcium, the cell membranes don't work right -- that is cells cannot receive the nutrients they need, not expel the waste they need to get rid of.
Normal water coming from a well, or from a stream, has leached an incredible about of calcium from the ground. As fishkeepers, we call this hardness. In most normal situations, the body is actually able to use a lot of this calcium from the water -- the gradient is favorable for calcium to leave the water and be put to use by the body. But, in RO water, since there is no minerals, this flow is reversed. The body is forced to give up calcium and loses the vital minerals it needs for survival. This is why re-mineralization of RO water is critical.
A secondary issue is how the fish does perform it's osmoregulatory processes. Salts are critical, since the fish performs a lot of ion-exchange at its gills. For example, in order to excrete ammonia, the fish usually excretes it as NH4+. In order to keep the charge balance, the fish takes back in another positive ion, normally Na+ is available in excess. Well, without any salts in the RO water, the fish is unable to exchange those ions. The result, the ammonia stays in the fish's body a lot longer than it normally would, the fish actually will begin to slow its metabolism down so that is doesn't produce as much ammonia waste and poison itself. Unless you are plan on inventing and using some sort of fish dialysis machine, using pure RO water and not allowing the fish to excrete its waste normally is going to cause long-term problems for the fish.
These are not immediate consequences. But, it is a long-term stress that is being forced on the fish's body. In an emergency, if you had to keep your fish in pure RO water, it would probably be fine for days, maybe even weeks, but it is not a good long-term solution.
Tap water does have a bunch of undesirable things is it, but tap water also has many good things. Not just calcium from my above example, but magnesium, manganese, etc. And don't forget the trace minerals, zinc, some iron, etc. The body doesn't need a lot of them, but so long as the concentration in the water was favorable, the body can use them when it needs it.
In summary, pure RO water is a very, very unnatural situation for a fish to be in, and pure RO water does not allow its body to perform its functions as designed. In analogy, consider being forced to live your entire life by breathing through a straw and the air is just a little low in oxygen. You can live for some time, but your body would much rather be able to take in large breaths with the right % of oxygen. So, remineralize that water -- a lot of people just mix in a percentage of tap water and that works pretty well. It isn't that hard or expensive, and your fish will live long and healthy lives with the remineralized waters.