DUBAI—Saudi Arabia and other Arab states boycotting Qatar have issued a list of severe demands to end the worst regional diplomatic crisis in years, telling their Persian Gulf neighbor to close state broadcaster Al Jazeera, curb ties with Iran and end Turkey’s military presence on its soil.

Qatar also must cut all ties with the Muslim Brotherhood, the Lebanese military and political movement Hezbollah and other groups that Saudi Arabia and its allies deem extremist and a threat to their rule, according to the 13-point list....