The ones I can think of are: allergies (other tenants might be allergic)noise (my bird is the noisy one in our household)property damage (yes, the little dog who is from California, pees in the house sometimes..ugh)Fear (my big dog is 110 pound black lab) of Dogs/Cats/AnimalsFear of lawsuits if the animal hurts someone elseI'd like to discuss what can be done to alleviate these and any other objections as much as possible.  

I don't/can't accept cats (allergies).

Pets certainly add more risk of renter to renter conflict, not to mention allergies, etc.

Sounds like it was a good standing prior; however, she may have some allergies unknown to her and very well can be mold in areas not visible to the eye.

You CAN refuse a therapy pet on the basis of lack of paperwork if its' purpose is "emotional support" or other not easily defined purpose.There is also a distinction now between "service animals" and "therapy animals" (I can't remember exactly what, but the laws are a little more forgiving for us for emotional support animals vs. true service animals) Yes, "no pets" is irrelevant although if you can make a case that an existing tenant has such allergies to that animal, that their life would be affected, it's possible to make a case.

Cat allergies are rare enough that it probably won’t affect the next tenant.

We stopped allowing cats after a couple of instances of later tenants having allergy issues. 

I.E unit b has a dander allergy and since the building is forced air you having a dog will cause discomfort?

Learn if they have allergies and avoid plants that trigger them. 

Harmless stuff unless you already have severe asthma, are of advanced age or have an allergy to it.