When you allow a tenant to move into your property you do so with the expectation that they will pay rent. The day that they stop paying rent that's owed for living there is a day that most landlords dread. When this happens (and especially if a tenant simply takes off without warning and leaves you with a unit full of their furniture and other belongings) it may be tempting to use anything left behind to help cover expenses that they've left you with, but you'll need to stop and make sure you're not breaking the law before either selling any of their property or simply tossing it in the garbage.
Only a few states allow a landlord to take possession of a tenant's property when that tenant gets behind on the rent or if damage caused to the property exceeds their deposit. This is called a 'landlord's lien' and is heavily documented and passes through a court who must allow it before the landlord can take action on his. Most states do not allow this and if a landlord attempts it they may actually be sued by the tenant. Suing the tenant or evicting them if they are behind on their rent is the option that most states require you to take.
If a tenant leaves behind property after they move out, local laws dictate what can be done with this. Some states, such as California, require the landlord to attempt to contact the tenant in writing and let them know about the property left behind. If the property is worth more than a certain amount, the landlord is required to auction it off if the tenant has not claimed it in the appropriate time period.
So remember, no matter what they leave behind or how far behind they are in the rent, you'll want to make sure you follow the law to the letter, otherwise you may find yourself with a bigger headache than you began with.