Can constant postnasal drip cause dry cough and chest problems?

Yes: It can aggravate asthma or lead to a cough by itself. Saline washes, nasal steroids plus an antihistamine work to slow or stop this.

Possibly: There are multiple causes of cough. The most common causes are post nasal drip, asthma, and reflux disease.

Postnasal Drip: Every day, glands in the linings of your nose make about a quart of mucus each day. The excess can come out of the nostrils -- that’s a runny nose. When the mucus runs down the back of your nose to your throat, it's called postnasal drip. http://www.webmd.com/allergies/postnasal-drip