My 1 year old has allergies but always has a runny nose regardless. Would this also be because of teething?

Not from teething. Teething causes drooling, fussiness, gumming hard objects. Speak with your pediatrician about an appropriate dose of antihistamine to give your child. If the runny nose is inadequately controlled consult an allergist.
Teething. Over centuries, teething has been blamed for fever, nasal congestion, runny nose and diarrhea, among other symptoms. It seems plausible that teething may cause irritability, night waking, drooling and finger chewing. It is easy to understand how the symptoms of a viral or bacterial infection were attributed to the teething process at a time when these germs were not yet discovered.