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As always, it is best to discuss your rights with a qualified attorney. Some of the information here may apply to you and some may not, depending largely on your state's own laws concerning emotional support animals. This article is meant as a general guide to some of the rights of landlords and as a starting point for your research on your own rights as a landlord, not as legal advice.
In order to avoid abuse of the system, landlords are generally permitted to require a letter from the tenant's doctor explaining the tenant is disabled by mental illness and how the emotional support animal is expected to mitigate this disability.
Under federal law, you do not have to permit emotional support animals for everyone with a mental illness, only those who are disabled and have a mental illness. The mental illness must be in some way directly related to the disability. This might vary with state laws, so check with your attorney. According to the NIMH, 26.2% of adults in the U.S. suffer from a mental illness in any given year, but only 6% are severely mentally ill. So more than three quarters of those with a diagnosed mental illness are not disabled by that illness and would not qualify to use a service animal even if they would benefit from one.
If the tenant has not yet gotten an emotional support animal, the landlord may be entitled to place restrictions on the size and breed of the animal. The landlord may be entitled to require the animal be spayed or neutered. Check with your state's Human Rights Commission or Attorney General for information on your state's laws concerning this. There are links to websites for every AG in the US in the service dog laws section of this site.
The landlord is not generally permitted to require a deposit for the emotional support animal, though there have been some cases where it has been permitted depending on local court rulings. Regardless, the landlord is permitted to charge reasonable fees to correct damage done by the emotional support animal once the damage has occurred.
If the dwelling has been modified to accommodate an emotional support animal, perhaps with a "doggie door" or fenced area of yard, and if the landlord desires it, the landlord can charge the tenant a reasonable fee for returning the property to the state it was in before granting the accommodation to have an emotional support animal. Generally the cost of any such modifications to the dwelling would be the responsibility of the tenant, not the landlord. This is not the same as the landlord being required to foot the costs to make facilities accessible to wheelchairs.
The landlord may be granted an exception if (a) his building has four or fewer units and the landlord resides in one of those units, or (b) the landlord is a private owner who does not own more than three single family houses, does not use real estate brokers or agents, and does not use discriminatory advertisements. Contact HUD or the Department of Justice for details.
The landlord may be able to have an emotional support animal removed if it poses a direct threat to others or disrupts the ability of other tenants to enjoy their dwelling. This is not a loophole to get rid of emotional support animals because someone doesn't like living in a complex with a pet next door. It is meant to give a legal remedy if the emotional support animal proves to be a nuisance, say by barking incessantly through the night so that other tenants cannot sleep peacefully in their own homes, or by the tenant refusing to clean up after their pet, posing a health risk to others.