@literally45-- That's a very good question. I read a study recently that was done on this topic. What they found was that a runny nose was not related to teething. Neither was a fever above 102 degrees.
The only symptoms they found associated with teething were irritability, drooling, itchy gums, biting, less appetite and a slight fever.
So if your daughter's runny nose doesn't go away, she must have picked up something. Watch for other signs of infection like high temperature and coughing and see a doctor if they appear. Teething never causes such extreme symptoms.