The British controlled Anglo-Iranian Oil Company was of immense importance for the British war effort. It was those Iranian oil fields that allowed the British navy to transform from a coal powered to an oil powered force. While Iran was neutral, the UK feared that the Shah held pro-German sentiments, which the British found unacceptable. The British were especially uncomfortable with the Shah's refusal to expel German nationals residing and working in Iran.
Once the Soviets entered the War, it became necessary to establish a rail route by which to supply the Soviets with war materials, and Iran was a prime location. These two factors prompted the invasion of Iran, and the replacement of Reza Shah with his son, Mohammed Reza Shah.
Interestingly, as an alternative to replacing Reza Shah with his son, Mohammad Reza Shah, the British considered replacing the entire Pahlavi dynasty, and returning the previous dynasty, the Qajars, under Hamid Hassan Mirza, to power. Hamid Hassan Mirza was a British citizen, and member of the former Qajar dynasty, who was serving in the British Navy at the time. The plan was abandoned once the British realized Hamid Hassan Mirza did not speak any Persian.
Source: Fakhreddin Azimi's 'The Quest for Democracy in Iran'
EDIT: I have mistakenly provided an answer to a question that OP didn't actually ask.
The city of Tehran was chosen for the meeting for diplomatic and geographic reasons. Also important, the Tehran Conference was the second of two meetings between the allied powers in a short period of time.
The first of the two was the Cairo Conference (which requires an explanation in order to understand Tehran), in Egypt. Cairo was chosen because, by 1943, Egypt was no longer under threat of Axis invasion. Also important, it was a geographic compromise in order to accommodate the attendance of Chiang Kai-shek, China's Head of State. Egypt was a nice middle point for Roosevelt, Churchill, and Chiang.
Stalin did not attend the Cairo conference. The reason for this was diplomatic. Although the USSR and Japan were both members of the Allies and the Axis, respectively, the two nations had signed a formal non-aggression pact in 1941. Japan was fighting a protracted war against Chiang in China and the Soviets were fighting Germans in the West, so it was in both their interests to maintain the non-aggression pact.
Because of this situation, it was impossible for Stalin to be seen in public solidarity at the same meeting with Chiang, Japan's enemy. Therefore, the Tehran Conference was chosen as the second meeting place in which Stalin, Churchill, and Roosevelt would be in attendance. Chiang would be absent. Tehran was a region of shared British and Soviet influence, and was relatively safe. What's more, it would have been very difficult for Stalin to fly West over mostly Axis territory to meet with his allies. Tehran, then, was a convenient, safe, southern route for Stalin that was quite close to Cairo, where Churchill and Roosevelt would disembark.
Chiang's absence at the Tehran conference would also result in several secret agreements among the other three allies concerning Soviet influence in post-war China. Chiang, who had always felt he was treated as an unequal ally, was enraged when FDR confided in him the details of the agreements some time later.
Source: Rana Mitter, Fogotten Ally: China's World War II, 1937-1945. New York: Houghton Mifflin Harcourt. 2013.