Born in Thurles, County Tipperary, he was educated in Thurles Christian Brothers' School, Coláiste na Rinne in Waterford, Copeswood College in Limerick, University College Dublin and the Law School of the Incorporated Law Society. He qualified as a lawyer and in the 1970s he worked with the Free Legal Aid Centres (FLAC), a voluntary group providing free legal aid, eventually became the chairman of FLAC.
In 1977 he was elected to the 14th Seanad Éireann on the Cultural and Educational Panel. At the 1981 general election he was returned to the 22nd Dáil as a TD for the Tipperary North constituency, where during the selection of candidates he had beaten future TD and Minister Michael Lowry for the Fine Gael nomination. He was re-elected at the February 1982 and November 1982 general elections. He supported the 1986 referendum to amend the Constitution of Ireland to remove the prohibition on divorce, and was denounced by priests in his constituency.
Molony did not contest the 1987 general election, and retired from political office to continue his law practice and to run his family's business in Thurles. However he remained involved in politics and was a director of elections for local Fine Gael candidates at the 2002 general election.
- "David Molony". Oireachtas Members Database. Retrieved 11 February 2013.
- "Death of Former Member: Expressions of Sympathy". Dáil Éireann – Volume 554. Houses of the Oireachtas. 9 October 2002. Archived from the original on 9 June 2011.
- "The salesman versus the idealist and the rest is history". Sunday Independent. 7 April 2013. Retrieved 14 April 2013.
- "David Molony". ElectionsIreland.org. Retrieved 11 February 2013.
- "Death of Former Members: Expressions of Sympathy". Seanad Éireann – Volume 170. Houses of the Oireachtas. 12 September 2002. Archived from the original on 21 August 2007.