I have both vented and unvented heaters in my house at present, and have found certain criticisms true, and others not to be, at least for me. Yes, they create moisture and carbon dioxide, and lower the oxygen level in the room, but if you follow the sizing recs of the manufacturer and the National Fuel Gas Code (which does allow them), these changes should not be significant. Also, many people who have problems with them do not allow the proper amount of fresh air to enter the heated space, which is clearly spelled out by the manufacturer.
On the other hand, they are 99+% efficient, and the extra moisture in the Winter, for me, is beneficial. They should not be giving off significant levels of any toxin if they are working properly.
My only complaint about them is that, even when they're working properly, to me they have a slight combustion smell. Apparently most people can't smell it, but I can. So I heat my basement and garage with them, to allow compliance with the air change requirements.
My sister heats her living room though with an unvented gas fireplace though, and would never give it up.
You can't vent ordinary gas appliances into a room though because they are not designed as stringently for complete combustion, and so they give off carbon monoxide, etc.
So, though people have strong opinions, I've never seen convincing evidence that unvented gas appliances cause any health problem, if they are properly installed and used. And if you have a real worry about excessive moisture, keep a simple hygrometer in the room to keep track of it. Again, if the unit sized properly and the air exchange rules are followed, moisture should not be a problem.
