Only if you add saltwater higher in salinity than your tank's would you increase the salinity. Ideally it would have the same salinity as your tank's. Your ATO replaces fresh water lost to evaporation, keeping salinity stable.
It would definitely take a while for salinity to change, depending on the total water volume. It happened to me, with a 180 gallon tank. I was thawing frozen food in fresh water and feeding with a turkey baster. So I was adding fresh water to the tank, in addition to what the ATO was adding. I started losing seagrass, and luckily found that salinity had drifted down too low. I corrected it and was able to save my seagrasses.
So it can happen. It's easy enough to check salinity occasionally.