This is a common question. In short, a landlord would likely be required to accept an animal used for therapy to assist an individual (tenant) in their disability or to assist in their use and enjoyment of the premises.
While you ask us to not confuse "Therapy Animal" with "Service Animal," under the Fair Housing Act they are really one and the same. The American's with Disabilities Act is a bit different.
The federal Fair Housing Act does not discern between the terms: "Service Animal," "Therapy Animal," "Emotional Support Animal" or other like terms. Where the law comes into play is where it says that you would need to "make a reasonable accommodation" to your rules and policies to allow a person with a disability to use and enjoy the premises. Courts have consistently ruled that allowing a tenant to have an animal to assist with a disability related a tenant's emotional state is, in fact, a reasonable accommodation.
There are multiply factors that are typically applied to be eligible for such a request:
- Is the person making the request disabled?
- Does the animal in some way help alleviate the effects of the disability?
- Is the request reasonable?
A tenant would have to meet all the above criteria.
There are certainly cases where 2 criteria may apply, however the request may still be unreasonable. For example, a tenant is requesting to have a Clydesdale horse in a 3rd floor unit of a downtown apartment complex. Are they disabled? Yes. Does the animal help? Yes (it helps pull them in a wheelchair). Is it a reasonable request? No (it would pose an immediate threat to the health or safety of others or the property).
Answering the question on whether or not the request is "reasonable" is honestly quite a gray area. It is best to look for cases of other similar instances or contact a local fair housing organization or real estate attorney if you are unsure. The United State Office of Housing and Urban Development (HUD) along with United States Department of Justice has put out a Joint Statement on Reasonable Accommodations under the Fair Housing Act that also helps to clarify what might or might not be a reasonable accommodation request.