TEHRAN—An ambitious hard-line cleric who is seen as close to Iran’s supreme leader will lead the judiciary as it investigates one of the country’s most revered and feared institutions, the Islamic Revolutionary Guard Corps, and its role in the downing of a passenger airliner last week.

Ebrahim Raisi, 59 years old, one of Iran’s most popular conservative figures, is considered a leading candidate to one day succeed the supreme leader. After losing the presidential election in 2017, he was appointed by Ayatollah Ali Khamenei...