I do know that in my Grey's former home her dust was an issue due to the owner's asthma or allergies. I don't think they had an air purifier; and as far as I know, they didn't have her for long (about a month).
I was never bothered by my Senegal's dander, and I've never been bothered by dog danger, either, but I do have dust mite allergies. That said, I've heard that it can be a good thing to get an air purifier when you have a parrot that produces more dust (e.g., Greys and 'Toos).
This is something I'm going to discuss with the avian vet who will surely make recommendations, but I'm curious of others' opinions for now.