SAQAR, Iraq—The artillery-scorched mountainside of this border area bears evidence of Iran’s effort to combat Kurdish rebels—just one of several groups who are now more vigorously fighting for greater autonomy within the Islamic Republic.

The Kurdish group operating here fled Iran more than two decades ago, but like other militants is now seeking to use international pressure on Tehran, particularly from the U.S. and its Gulf allies, to advance its struggle.

With...