Born in Sjernarøy, Furre moved to Stavanger in 1946, and then to Oslo in 1947. In 1959 he became the leader of Det Norske Studentersamfund. He was a member of the Labour Party, but as the chairman of Sosialistisk Studentlag (1956-1957) he was central to the so-called "Easter Uprising" in 1958. He later left the Labour Party to found the Socialist Left Party together with other prominent persons in the Easter Uprising, such as Finn Gustavsen.
Furre graduated with a cand.philol. degree in 1968. He was a professor of history at the University of Tromsø between 1985 and 1986, having worked there since 1971. From 1991 to 2007 he was a professor of church history at the University of Oslo. He was ordained a priest in 1998, 42 years after he had started his theological education.
He has been a member of the Norwegian Nobel Committee since 2003. He was a member of the Lund commission 1994–1996. During this period, Furre himself was secretly investigated; when this came to light, it prompted the previous Minister of Justice and the Police Grete Faremo to resign from her then post of Minister of Energy.
Furre has written a number of books, particularly on historical and political topics. Furre's book: Norsk historie 1905-1990: vårt hundreår, is used as a general introduction to Norwegian history at several universities and university colleges. He was the editor-in-chief of Syn og Segn between 1967-1971, and the vice chairman of Noregs Mållag between 1960-1970.
- 1990 Soga om Lars Oftedal ISBN 82-521-3468-8 - about the clergyman Lars Oftedal
- 1992 Norsk historie 1905-1990 : vårt hundreår (History of Norway 1905-1990 : our century) ISBN 82-521-3487-4
- 2000 Norsk historie 1914-2000 Industrisamfunnet - frå vokstervisse til framtidstvil (History of Norway 1914-2000) ISBN 82-521-5187-6
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