Yes, even Islamic theocratic republics have them:

Like many online scandals in the West, it involves a model. Not Paris Hilton, but a supposed model of virtue: a cleric.

In the video—for weeks voted the top story on Balatarin.com (an Iranian version of Digg.com)—a robed cleric is caught on a hidden camera in a private room. He walks to the door to let a chador-clad woman enter. “Nobody saw you come in, did they?” he asks her lightheartedly. As she removes her chador, he continues in the same tone: “Want to do some Nasnas?”

Iranians know Nasnas as a mythological monster. What the cleric means by “do some Nasnas” is clarified by what happens next in the clip. Americans have a similar expression: the beast with two backs.

For an even bigger sex-tape scandal from Iran, read this Wikipedia page.