Yes, you can have a chronic runny nose from a moth ball allergy. An allergy to moth balls can effect the entire respiratory system in persons who are more susceptible such as people with asthma or COPD.
The symptoms are that of any other allergic rhinitis: sneezing, itchy, runny nose and watery, itchy, red eyes. Of course people have allergies in varying degrees. You may not have all of the symptoms to be bothered by moth balls.
There are plenty of other particles in the air to also cause allergy symptoms. Here is a good, easy to read resource on dealing with environmental allergies: http://dsa.csupomona.edu/shs/files/Allergies_1.pdf
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