He was born in Tbilisi, Georgia (then part of Imperial Russia), to Prince George Bagration of Mukhrani (1884-1957) and his wife Helena née Zlotnicka. The 1921 Red Army invasion of Georgia forced the family to leave Georgia. Educated in Germany, Prince Irakli then settled in Italy in the 1930s and married Maria Antonietta née Pasquini dei Conti di Costafiorita in 1940. Following her death in 1944, Irakli, with his infant son Giorgi, moved to Spain, where he naturalized and married Infanta Doña María de las Mercedes Raimunda (1911-1953), a niece of Alfonso XIII of Spain, in 1946. She died in 1953, leaving the daughter Mariam (born 1947) and son Bagrat (born 1949) behind, and Prince Irakli married Doña María del Pilar Pascual y Roig, Marquesa de Carsani, in 1961.
Irakli Bagration-Mukhraneli played a prominent role in Georgian political emigration and, as an active royalist, remained in opposition to the Soviet rule in Georgia. He founded the Order of the Eagle of Georgia and of the Tunic of Our Lord Jesus Christ in 1939 and sponsored the establishment of the Germany-based Union of Georgian Traditionalists, which fought to restore Georgia’s sovereignty from the USSR and advocated a constitutional monarchy as a form of the government for independent Georgia. Following his father’s death in 1957, Prince Irakli succeeded as Head of the Princely House of Mukhrani and declared himself Head of the Royal House of Georgia, assuming the style of His Royal Highness. He died in Madrid in 1977. His remains were reburied to the Svetitskhoveli Cathedral in Georgia in 1994.