Molony was born on 27 February 1877 at Ennis in County Clare, Ireland to James Barry Molony and Ellen Greene.
Molony passed the Indian civil service examination in 1899.
From 1908 to 1910, Molony served as Assistant Political Agent to the state of Banganapalle.
Molony served as president and Mayor of the Madras Corporation from 1914 to 1919 and City Commissioner in 1920.
Molony retired from the Indian civil service in 1925 and returned to the United Kingdom Molony died on 5 September 1948 at the age of 71.