I have been in a few homes with ventless fireplaces and immediately noticed that extra moisture. Lots of water vapor produced burning NG and it's as bad for your home is excess CO would be for your body. I'm assuming it's an older home 15+ years or more. This is good in that the home is probably leaky and let's you get some air exchange. Newer homes are much tighter and many locations require ACH50 of 3 or less now. (Air changes per hour at 50 pascals) many older homes double or triple that.
The homes I have been in with ventless fireplaces remind me of the feel of my Grandmother's house that was heated with those wall mounted propane heaters. It was in the 90's when we gutted that house and found huge mold growths in the areas around the two heaters. I was a teenager at the time, but mold always meant water leaking in from the outside to my dad and uncles.
Now 25 years later and having gone through lots of building science education, I know that most mold is caused by interior moisture that can't get out. Those propane heaters came out of her house because of central heat and air, but I guarantee you that it added years to her life. She's currently 89 and still mean as a hornet.
Ventless fireplaces would be especially bad in the south. Southern homes are typically built to keep high humidity out since it's hot and humid outside and dryer inside most of the time. In colder climates, it's the opposite with cold winters and gas furnaces. Adding high levels of humidity into a home built in the south seems like a good way to grow a lot of mold.
For my take I would look to find another way to heat the house. Whether you have a gas or all electric hvac system, the monthly energy expense pails in comparison to the costs of mold remediation not to mention the potential health hazard of mold and CO.