Other than the potential for physical property damage caused by the pet, what kind of liabilities does the landlord open themselves up to when allowing tenants with pets?  

Make sure they will do the credit check, background check, show the property, generate the lease, discuss up front what credit requirements you require, what pets you will accept, etc.  

Also, people with pets usually post pics of their furry friends online, informing a landlord if the potential tenants are being honest when they say they have no pets.

Maybe having business cards on you at all times and handing them out like candy, or telling everyone you know and their pets that you buy houses.

@Valerie King , the standard lease agreement doesn't really have much in it, other than how much they will pay and what the deposit will be.It doesn't address whether or not you allow smoking, pets, who shovels the snow, where they can and can't park, etc.All these things are not covered in the Landlord Tenant Laws of your state, which are very important to read on their own, but just a beginning.I suggest you do a search on BiggerPockets for Lease Clauses - you'll see some pretty amazing clauses here from landlords who have been doing this since before dirt was invented.Marcia Maynard has one of my all-time favorite lease clauses.

That could mean allowing pets or allowing non-violent felons.

Section 1941 also states that the landlord must repair problems that make the property uninhabitable – except for problems caused by the tenant or the tenant's guests, children or pets.

6.They allow pets without your consent.

Ask for the pet deposit per the signed lease .

They also will not make you whole for any pet damage whatsoever.