Irani was born in Lebanon and is of Palestinian origin. He received a bachelor of science degree in chemistry at the American University of Beirut in 1953. He moved in the same year to Los Angeles, California, and at age 18, began graduate studies in physical chemistry at the University of Southern California. He received his PhD in 1957. He subsequently worked as a researcher for the Monsanto Company until 1967 and joined the Shamrock Corporation. Prior to working for Occidental, he was president and chief operating officer of Olin Corporation, a chemicals and metals company.