Question

What tensions between the Allies were revealed at the conferences in Casablanca, Cairo, and Tehran?

Answer 1

A primary tension between the Allies was tension between the Western partners (the USA, Britain, and France) over against the Eastern powers (the USSR and China) - and there was tension between the USSR and China as well.  There were tensions about how war ends would be pursued.  The USSR under Josef Stalin particularly wanted assurances that the war would be fought until an unconditional surrender by both Germany and Japan. Stalin also wanted a second front to be opened in the war in Europe, to relieve pressure on the Eastern front where Germany was battling the USSR.
Roosevelt, Churchill and DeGaulle (representing the US, Britain, and France) met at Casablanca in January, 1943.  Stalin was invited but did not attend due to the difficult state of the war in the USSR at that time.  They promised to fight on to the Axis Powers' unconditional surrender. They also discussed opening a second front in Western Europe, but did not determine a specific plan.
Roosevelt, Churchill, and Chiang Kai-shek of China met in Cairo in November, 1943, focused particularly on dealing with Japan and the future status of Korea.  Stalin had refused to attend this conference because of China's participation.  (Those two nations were rivals to one another.)
Roosevelt, Churchill and Stalin met in Tehran in November, 1943, just days after the close of the Cairo Conference.  Plans for an invasion into France were discussed, to open up a Western front in the European theater of war.  This would be Operation Overlord, which we now typically refer to as the "D-Day" invasion at Normandy.