There is really not a lot of history related to our (USA) activities in Persia during WW II. This was primarily the concern of the British and Russians, who were allies in the matter, and who had keen interest in the oil-rich country.
It turns out that both the British and Russians suspected the Shaw of being an Axis sympathizer.
After Germany invaded Russia in June 1941, the two powers agreed on the joint occupation of the country in order to protect their interests, which was accomplished in August 1941. There was no German or Persian resistance to the action.
In January 1942 they signed a treaty with Persia, guaranteeing the return of the country to absolute Persian rule at the end of the second world war.
Our involvement was limited to our approval of the treaty in December 1943, at the Tehran Conference.
True to the treaty, the British and Russians withdrew from the country when the war ended.
Persia was a pivital route for military supplies sent from the west allies to the Soviet Union in the course of the war.
That's about all I know on the subject. Hope it helps.