What are some quick procedures that can successfully remove pet dander?

Bathing: If you can do nothing else, bathing the cat or dog, ideally twice per week, will help the most. Hard floors, vacuuming with a filter and using a hepa filter in the room will also help. Nothing eliminates pet dander though, other than relocating the animal plus time and cleaning.

Real estate agent: Find one. Buy new house. Leave pet behind. There are no other "quick procedures". Moving pet out is your best solution in long run. Keeping pet off furniture, bedding & out of laundry room helps. A hepa air purifier reduces airborne pet dander as will keeping windows open. For effective hepa calculate volume of home. Hepas must move that amount of air every 10 minutes. It's a big investment.