How to kill fleas on a finch without harming it?

Your Finch does not have fleas. They do not get on birds. If you had described what is going on, it might have helped me to at least give you a guess. Whatever you do, do not put anything on your bird and do not waste your money on any of those so called mite protectors from pet stores. They can poison your bird. Make sure there is enough humidity in the area and make sure the bird has plenty of opportunity to take a bath. You can offer it a pie pan with warm water for bathing. Without more info I can't be much more help than that.