The good news is human colds are not transmittable to rabbits. However, there are a variety of reasons your rabbit may have a runny nose and many may require veterinary attention.
One common cause of a runny nose is a respiratory infection. An upper respiratory infection in rabbits is commonly referred to as “snuffles.” It can be caused by a variety of bacteria, so it is best to discuss with your veterinarian the best antibiotics to use to treat snuffles. Lower respiratory infections (pneumonia) are often associated with difficulty breathing and require immediate veterinary care.
Dental problems such as overgrown teeth can cause a runny nose in rabbits as well. This is due to the fact that the teeth are located directly under the sinuses, and tooth or root overgrowth can cause blockages of the tear ducts.
Finally, there may be a foreign body such as hay or bedding that has become lodged in your rabbit’s sinuses. This may be something you can see and remove yourself, or it may require veterinary care.
For more information on upper respiratory infections and dental issues, see these articles: