"I have a request for housing from someone with a Therapy/Emotional Support animal (cat). Our company offers furnished and unfurnished apartments. . . Is it lawful for me to request that she live in one of our unfurnished units so that her cat is not damaging or getting hair on our furnishings (couch, recliner, beds, etc.)?"
These type of situational questions get very difficult to provide any concrete advice to. Are the current unfurnished units subject to the Fair Housing Act? Would they fall under transient occupancy laws? What are the lease lengths?
Typically a concern about hair or damage is not a legitimate excuse to not allow a disabled person to have a reasonable accommodation (cat). This would be akin to telling someone in a wheelchair that can't rent the nicer unit because the floors might get scratched by the wheel chair.
I can tell you if there is damage then you could require that the tenant pay to correct it and return the property to the same state it was in at the commencement of the lease.
As with any difficult fair housing question it may be best to contact an attorney or fair housing agency for further assistance.
For more information and resources on reasonable accommodations please review this post.