There are many reasons to not allow pets, and some depends on your properties and furnishings.
(1) Property Destruction - Pets do cause damages, and sometimes major ones. Especially a pet in a new environment like just moving to a new home, is most prone to cause damages. I have had dogs pulled off kitchen cabinet doors, bite off corners of solid wood doors, dug big craters in the ground outside, broke sprinkler risers, broke sprinkler control wires, chewed off fence pickets at the bottom of wood gates to get outside, cats tend to completely crawl off weather seals around doors, and destroy window blinds, and carpets. If you have a furnished property then anything with fabric is a problem.
(2) Litter management - this is not the fault of the pets, but the pet owners. Tenants that let their dogs do their #2 in the yard, creating land mines they do not pick up. Have you ever had your own lawn maintenance call you and say that there are so many mines in the yard when they mow it's literally slinging poop all over. Embarrassing to have your neighbors call you and tell you your tenant just left a big pile on their front yard. Or have tenant goes on vacation for a week, leaving a seven day worth of a giant pile of wet food on a bowl set on the floor, attract ants and roaches, which resulted in an infestation? I have even heard of tenants flushing cat litter down the toilets, not knowing that they swell up like crazy to seal the drain shut. You end up with many additional headaches that result in your tenants not being good responsible pet owners.
(3) Insurance Paperwork - many property insurance companies now do unannounced outside inspection. They look around for trampolines, above ground pools that's not fenced in, fire pits, tiki torches and any other liability concerns. I have received letters from the insurance companies, on a property I bought and have inherited tenants with dogs. The inspector cited dog presence, and the insurance company wanted wellness documents, breed and size, and bite history etc etc etc...or they would not renew my policy.
(4) Maintenance Around Pets - Ever having to enter a property to repair something and having to deal with dogs snarling, barking, baring their teeth blocking your way? It's more work when making repairs with pets around. I once had an internet provider enter a property and his knee pads were crawled by a dog then he refused to enter. Not only you have to worry about access, you have to be extra careful to not let a dog or cat out while you carry tools and parts into a property. If you are doing the work alone, try to think how you replace a 40 gallon water heater with a dog trying to sneak out every time you open the door. Yes you can ask the tenant to keep the dog away, or keep them in a room with door closed, but that's not always practical if they have to work that day. Or half way through your repair, you need to go get an extra part, what do you do with the pile of screws on the countertop, or the drill, or a razor blade on the floor? You can't leave them like that if there are pets around, you need to put that all away, while you are getting the parts and reverse after you come back, because what if a cat some swallow a few screws? Or squeezed him/herself into the wall cavity you just opened? Once you are done, say patching some epoxy on the window sill, or painting a wall, or baseboards...how do you prevent the chemicals from being licked or scratched? You need to move something in front of it, or work on a perimeter jig while the work cures or set. It's is extra work.
(5) Wear and Tear - This is not damages, but pets cause a higher level of maintenance. Ever looked at how dirty your AC filters are with pets and compared them to the same without pets? The pet occupied property filters are always black, and the no pet property filter light gray. All the hairs gets onto the filters and beyond. They go to your air handler coils, and duct work. I have a fourplex which I replaced all the AC units at the same time, and I was able to compare the units with pets and without pets. With no exception, the pet occupied units get much dirtier filters (which I personally change once every 30 days), and the coils needing acid washing much more frequently. Your air conditioning units will have a shorter life span because of furry pets.
It's much more than just property destruction.
