Symbols are objects that have a deeper meaning than their literal meaning. In Fortunato's case, a cough is more than a cough. It symbolizes two aspects of his situation beyond merely his being a man with weak lungs who coughs when he is in damp, moldy spaces.
First, it symbolizes his drive and ambition. When he pursues a goal, he doesn't notice seemingly petty distractions or let them get in his way. He repeatedly dismisses his cough as of no consequence in comparison to getting the chance to sample the amontillado. This characteristic can be important for getting ahead in life, but the single-mindedness it represents also gets in the way of Fortunato perceiving all the warning signs of danger that might have alerted him that all was not well (along with the fact his judgement is impaired because he's been drinking). He downplays his vulnerabilities to his own peril.
Second, it symbolizes his vulnerability, and in a larger sense, the vulnerability of the human body. Fortunato is another flawed human like the rest of us, and his cough, in a visceral way, raises our sympathy for his plight. He will die suffering, and that is a cruel fate.