If the animal is disturbing neighbours, the tenant might prevent the animal from being outside and limit its exposure to neighbours, if the animal is upsetting a fellow tenant's allergies due to fur contaminating the communal laundry machine, the pet owner can do their laundry elsewhere.

They then started to complain allergies/asthma and they said they suspected mold, although no mold could be found in the living area. 

Virginia is a caveat emptor state, however you can not hide latent defects, and doing so with the mold could jeapordoze someones life and cause death if they have a mold allergy. 

https://www.kcrg.com/2020/07/02/iowa-supreme-court-sides-with-pet-allergies-over-support-animal-in-tenant-dispute/This is an unusual case, but I bring it up because it's likely to become more common as emotional support animals become more common.Read the article all the way through.

Our tenant of 2 months is complaining about allergies that he says seem to be triggered by having the HVAC on.

The court said the landlord should have denied the ESA since it created a medical issue (allergies) for an existing tenant.

Nowadays many tenants have allergies and won't rent a unit that has carpet.

Volunteers are being solicited by the National Institute of Allergy and Infectious Diseases.

He stated that his wife has pet allergies and needed clean carpets, or he would have no choice but to replace the carpets.  

Part of the reason I was moving from Oregon to Nevada was to escape allergies.