Nitromethane does not contain nitric acid, nitro methane's chemical formula is CH3NO2. Nitric acid is a byproduct of combustion, it's chemical formula is HNO3. Water (H2O) and Carbon Dioxide (CO2) are also produced, along with a bunch of other compounds when mixed with methanol like racing nitro is.
So now to the original question. Does nitro burn your eyes more on humid days versus dry days? There is obviously going to be more water vapor in the air and therefor more Hydrogen and Oxygen for the Nitrogen to bond to during the combustion process in high humidity areas. The air that we breathe, the Earth's atmosphere, is also between 67-70% Nitrogen, which would lead to more Nitrogen bonding with more Hydrogen and Oxygen during combustion in high humidity areas. Without getting a chemical engineer involved, I am going to say YES, there will be more nitric acid produced at St Louis or Atlanta than there will be at Vegas or Phoenix. Is it a lot more? Probably not, but more is more, so it will be more of an irritant.