Can constant coughing damage the lungs and affect lung function?

It would not be from coughing per se, the reason that is inducing the cough could damage the lung. Any cough lasting longer than 2 weeks warrants an investigation by a lung doctor/Pulmonologist.

Chronic cough has to be investigated and treated, otherwise not likely to cause lung damage by itself. Cough is a symptom triggered by various causes, lung and not lung related. Keeping a log of cough spells may help in better identifying a possible cause.