What are the signs of a tooth infection does it make a person cough up mucus?

No corelation. The symptoms of a tooth or gum infection are usually pain and swelling. There would be no reason that coughing up mucous would be tooth related.
Not usually. In fact in my 30 years of dentistry I have never seen this happen. However, in otday's world nothing would surprise me. See your dentist immediately if this is what you suspect is going on in your mouth.
Pain, swelling. Dental infections usually cause tooth pain or swelling. Sometimes, pain can be in the upper cheek. This is also the location for a sinus infection, which is more likely to cause mucous. You should visit your doctor for an examination to determine your diagnosis.