Germany used its naval fleet to stop allied arctic supply route to the USSR. Were there any German operations, in Iran and Iraq (who were pro-German) to throttle allied supply routes through the Persian corridor? They could have easily found operatives to work for them there.
Donal O'Sullivan's Dealing with the Devil: Anglo-Soviet Intelligence Cooperation During the Second World War (2010) looks at the intelligence situation in Iran:
Pro-German sympathies within the Iranian population might fluctuate with the war events, but the Allies managed to prevent major subversive activities. They captured Franz Mayr, the Chief German SD representative in August 1943 as well as Berthold Schulze-Holthus, and experienced Abwehr officer, in March 1944. The British captured Mayr together with four wireless operators, who had been dropped by parachute.
Surprised by the Allied invasion of Iran in 1941 and woefully unprepared for their tasks, the German spies did not even have a regular courier service to Berlin. In any case, both officers refused to organize sabotage, fearing it would quickly lead to their capture. Instead, they concentrated on establishing a network of dissatisfied Iranians and tribal leaders for a future revolt. Mayr and Schulze had been productive in recruiting Iranians for underground work, but the combined efforts of Soviet and British intelligence thwarted their efforts.
He does not buy into historical rumors about the planned assassination of Winston Churchill, Franklin Roosevelt, and Joseph Stalin at the 1943 Tehran Conference (Operation Long Jump, claims of which may have been a Soviet propaganda effort). Operation François also in 1943, a German parachute mission to make contact with the dissident mountain tribes to encourage them to sabotage Allied supplies, seems to be better sourced, if also ineffective.
German radio propaganda was broadcasted e.g. from Athens. I don't know how far it reached and whether it may have called for sabotage next to the obvious anti-Semitic agitation.
Overall, it seems at least in Iran there were neither direct nor effective (from a Nazi German point of view) operations to destroy supply routes in the Persian corridor in World War II.
Yes U-Flotte 33 was deployed in several waves from 1943-44 to intercept Allied shipping in the Indian Ocean. The earliest of these deployments in 1943 suffered from lack of range by the Type-IXC boats used but these were superseded by longer range Type-IX-D2 boats of the Monsun Gruppe missions.
Some of these like U-198, U-861, U-862, U-859 and U-852 were specifically tasked into the Gulf of Aden to intercept US special convoys 300 and 301 to Persia (Iran).
Special Convoy 300 comprised 12 Freighters which left New York July 3rd, 1944 including the vessels Benjamin Bourn, Sydney Sherman and John Barry. U-859 torpedoed John Barry on 28th August 1944 off the coast of Oman. Sydney Sherman was surprised to be bombed by long range German aircraft.
There were some quite pointed allegations made by US Consul William Eddy accusing that Churchill had deliberately leaked information of these convoys to Hitler via back channels to ensure these two convoys were stopped.
The United States in turn was accused of smuggling nuclear materials to the Soviet Union under Lend Lease with these convoys contrary to a prohibition Churchill had imposed upon the use of Tube Alloy secrets and Canadian Uranium.
The overall capability of the Kreigsmarine to deter shipping to Persia was extremely weak. Less than a quarter of U-boats dispatched to the Indian Ocean survived. Most of the total 47 U-boats sent to the far east were sunk outbound from France in 1944.
There was at least preliminary steps by the Germans to use the Vichy French-controlled Syrian Republic and the French Lebanon as a springboard to occupy Iraq. Vichy France had forces in Syria and from 1 April 1941, after a coup d'état, Iraq was controlled by pro-German rebel forces under Rashid Ali. Vichy France was also increasingly cooperative with Germany. There were indications that actual German forces were beginning to move in.
Great Britain, fighting in North Africa, couldn't allow this to stand for a number of reasons - the threat to Egypt and Mideast Oil, to name two. So a scratch force was sent north to Syria : "Operation Exporter", that, after some difficulty managed to capture and occupy the region and end this threat of a land route to the Mideast.