There is no exact correlation between the oil industry and Islamic extremism, but one can make a stretch that one has an impact on the other. In political economics, there is a concept called the "Resource Curse". As I am more familiar with Iranian history, especially during the end of Mohammad Reza Shah Pahlavi's rule, I will use him as an example.

Prior to the 1973 oil crisis in the United States and Europe, Iranian oil was closely regulated, or rather monopolized by the Anglo-Persian Oil Company (APOC). Once word came out that the American-owned Arabian American Oil Company (Aramco) agreed to share 50-50 of the profit with the Saudis, Iranians became more aware of the monopoly the British had on Iranian oil, a 75-25 ratio in favor of the British. Although Mohammad Mossadegh failed to nationalize APOC because of the coup d'etat in 1953, it set a precedent for both the rise of Mohammad Reza Shah and the gradual integration of sharing the oil wealth with the Shah and the west.

The 1973 oil crisis became a turning point for both the West and Iran. Western nations, which faced one of the worse economic recessions since the Great Depression, were paying oil per barrel triple or even quadruple the rate pre-1973. Each barrel of oil had risen from 90 cents in 1970 to $7 by 1974. This translated into a jump in net profit from $4 billion to $60 billion over four years.^1 Iran suddenly faced a huge bloom in its economy. Iran's GNP rose by 8 percent per annum in the 1960s, 14.2 percent in 1972-3, 30.3 percent in 1973-4, and 42 percent in 1974-5. The GNP per capita has also risen substantially from around $450 in 1971 to $2,400 in 1978.^2 See figure 1 ^3 Iran's oil revenue tripled after 1973, but this is where the Resource Curse comes in.

The increase revenue and reliance on oil only exacerbated the wealth gap in Iran, the rich were getting richer, and the poor were left out of the oil bloom. The Shah had disillusions of grandeur, where he started projects that costed too much money or corruption left some projects in ruin. One of those programs was creating Nuclear energy for peaceful purposes, which its controversy still exists today. Iran's military expenditure also dramatically increased. In figure 2^4 the amount of money spent and the complete reliance on the U.S. alienated ordinary Iranians and fueled Islamic extremism. Hence, many of Ayatollah Khomeini's initial disagreements with the Shah was his wealth and western influence, which both came from the oil revenue post-1973.

TL;DR: Iran Shah, Mohammad Reza Shah Pahlavi, exploited oil revenues post-1973 to build up his "empire", but it only fueled opposition among the people, especially with Ayatollah Khomeini who is seen as the founder of Shia extremism.

1. Farmānfarmāʼiyān, Manūchihr, and Roxane Farmanfarmaian. Blood and Oil: Memoirs of a Persian Prince. New York: Random House, 1997.

2. Halliday, Fred. Iran, Dictatorship and Development. Harmondsworth: Penguin, 1979. Pg. 138.

3. The Middle East and North Africa, 1976-1977. p. 94 (for 1966-75).

4. Michael Klare, based on U.S. Department of Defense publications.