Under a variety of circumstances, a landlord may be held liable for bites by tenants' dogs. A dog bite victim can sue a landowner or property manager if there was a vicious dog living on the premises, a defect like a broken gate that allows a dog to escape and injure someone, or a broken promise to do or not do something like fix a fence or prohibit pitbulls. 

A vicious dog is dangerous, as is any dog that gets loose and runs around the neighborhood. Landlords have a legal duty to inspect their property and fix or get rid of dangers. 

Attorney Kenneth M. Phillips recently won a case against a landowner and property management company in Stockton, California, after a mother of three was killed by a pitbull. The owner of the pitbull had been evicted but was living in a shed on the defendants' premises, and the driveway gate could not be secured shut, allowing the dog to attack the woman. Phillips based the lawsuit on the defendants' failure to fix the gate. He proved that the defendants never inspected the gate despite knowing there were medium-sized dogs on the property.

The only exception is a California residential landlord who doesn't know about a vicious dog because it was not there at the beginning of the lease and was never brought to the landlord's attention afterwards. This exception applies only in California.

Phillips recommends that landlords require tenants to maintain a policy of renters insurance that covers their dogs and has a limit of at least $100,000. He also suggests using a "dog clause" in leases that would lower the risk of liability by negating the factors that these lawsuits are based on. A comprehensive one can be downloaded at this website

For further information:

Liability for bites by tenants' dogs upon property owned by the landlord
Liability for bites upon adjacent property
Recreational immunity
Further research