A runny nose can be from any sort of irritant--dust, pollen, etc or it can be due to a bacterial or viral infection. Viral infections especially can last for weeks. Bacterial infections tend to cause a colored nasal discharge while viral tend to cause clear or slightly white nasal discharge. You can also have a viral infection with a secondary bacterial infection.
A persistant runny nose or one accompanied by lethargy, change in appetite, coughing, fever, etc would be reason for a vet exam.
Licensed Veterinary Technician