What is the point of a runny nose during a cold?
Edited by request for clarity

The biological purpose of a runny nose is to clear pathogens and allergens trapped in the mucus out of the nasal cavity and sinuses as a result of rhinitis or sinusitis. 


Mucus is considered to be part of the innate immune system and provides mechanical and chemical defenses against pathogens.


Mucus acts as a barrier protection that is difficult for pathogens to breach.
Mucus is capable of trapping pathogens in it and when that mucus is swept out of your upper respiratory track through your nose or down your esophagus, the pathogens are removed from those areas as well.
Defensive proteins called Mucins, Lysozymes, and Immunoglobulins are secreted into the mucus and provide additional protection, as they can lyse or opsonize pathogens.

Mucus keeps epithelial cells moist so that they do not dehydrate as air passes over them. This in turn keeps epithelial cells healthy so that they maintain a tight cell layer in order to protect the underlying tissue.


As part of the mechanical defenses against pathogens, epithelial cells are ciliated and they rhythmically sweep mucus from the sinuses and nasal cavity mainly into your esophagus and digestive system, but can also move mucus into your nose. Most of the time we aren't even aware this clearance of mucus is happening and it travels down our esophagus into our stomach's and intestines where it can be excreted from the body. This is the case when you do not have an infection and your mucus is not viscous.

When you have an infection the inflammatory response is triggered by cytokines from macrophages. Those cytokines induce cells in the area to do many things. One is to increase mucus production. This may cause epithelial cells in your nostrils and nasal cavity and sinuses to secrete more mucus as well, and this will be swept away through your nose, where you will notice it as a runny nose. 

Usually your nose is runnier with rhinovirus infections or with allergens than it is when you have a bacterial infection where mucous tends to be thicker and more viscous. The ciliated epithelial cells have an easier time moving the thinner mucous of a viral infection or allergy than the thicker mucus of a bacterial infection, so you notice a runny nose more when you have a cold or hay fever.

So in short, the biological purpose of a runny nose is that it is part of the immune response to clear pathogens away and protect your uninfected epithelial cells from being infected. It also clears out much of the debris and inflammatory proteins that are found at the site of the infection so that they do not remain in the area to further inflame tissue that is trying to heal. When you have allergies your immune system reacts inappropriately to a benign substance, and if the allergen finds its way into your nasal cavity, it will generate rhinitis and cause your nose to run.