What happened to democracy in Iran? It didn't die a natural death. Iranian democracy was killed by well known actors that included British Petroleum and the United States. You can learn more by listening to Stephen Kinzer, former New York Times reporter, who was interviewed by Amy Goodman of Democracy Now (video interview). From the 1920's through the 1940's Great Britain controlled all of the oil in Iran, thanks to "a corrupt deal that they had struck with a few representatives of the old declining Iranian monarchy, all of whom had been paid off." The troubles began when Iran began to assert ownership of its own oil:

[A]fter World War II, when the winds of nationalism and anti-colonialism were blowing throughout the developing world, Iranians developed this idea: we’ve got to take our oil back. And that was the general—the kind of national passion that brought to power Mohammad Mosaddegh, who was the most prominent figure in the democratic period of Iran during the late '40s and early ’50s. It was Mosaddegh's desire, supported by a unanimous vote of the democratically elected parliament of Iran, to nationalize what was then the Anglo-Iranian Oil Company. They carried out the nationalization. The British and their partners in the United States fiercely resisted this. And when they were unable to prevent it from happening, they organized the overthrow of Mosaddegh in 1953. So that overthrow not only produced the end of the Mosaddegh government, but the end of democracy in Iran, and that set off all these other following consequences. The Shah ruled for twenty-five years with increasing repression. His rule produced the explosion of the late '70s that produced the Islamic regime. So, it was to protect the interests of the oil company we now know as BP that the CIA and the British Secret Service joined together to overthrow the democratic government in Iran and produce all the consequences we've seen in Iran over the last half-century.