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:
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.