I have used Eucalyptus Oil, Rose Oil, Citronella Oil and Lavender Oil in an oil burner in the same room that my birds were in and have had not problems. Budgies nest and live in Eucalyptus trees chewing the wood and leaves with no harm to them. Birds can eat Roses safely. Citronella is a natural insecticide as is Lavender Oil.
But others may have different opinions and proof that these oils are harmful. I have never tried Peppermint Oil. The perfumes used in air fresheners etc are artificial and usually do not come from natural sources as well as the other chemicals that they contain.