It depends. We lived in a place where we refinished the floors and on hot humid days we could still pick up a faint smell of cat pee where they had the litter box. Not terrible but we could still detect it slightly.

Dog smell is probably easier to eliminate.

I would rule out any home that you have a reaction to when you are in it. Maybe not all homes with cats or dogs will bother you. Start to visit different homes and find out. Do you have a reaction when you go to a friend or relative who has pets? Some pet houses smell much worse than others.

I am insanely allergic to plug-in air fresheners and cigarette smoke. I ruled out all homes of smokers and homes that used plug-ins. If I had a serious reaction within minutes of being in the house, I knew that it would be very hard to get the smell out of the home because I'm that sensitive. If someone is a smoker but not in their house, I still get sick from being in their house because the smell is on their body and their clothing. It narrowed my list down but I never had to pass up a dream home because of it.

Luckily, we live in an area with very few smokers. That would be a hard pass for me. Yes, my sister used to have a cat and my eyes would get itchy in their house. My real problem is if the cat sits on me or rubs against me.

We had a poodle in the past and I don't think our house smelled. I also could have been nose blind!

You left the open house with itchy eyes and a headache. That's an allergic reaction and more apt to be caused by rugs and fabric. You didn't complain about odor. House is emptied, hire a good cleaning company to go in and clean top to bottom.

Smoking? Now that an issue for me as it has contaminated every inch of the home. Don't you just love nicotene yellow walls and ceilings?