Well, many today call Hull Presidency a "FDR Presidency in all but name" as Cordell Hull and FDR switched positions with Hull President and Roosevelt new Secretary of State. Roosevelt remained pretty popular and was selected by Hull as the first General Secretary of United Nations in 1946, winning a Nobel Peace Prize.
Probably with a third term he would be perceived as too partial for the role and would be replaced by someone other (Hull himself maybe?). Be President is also a very stressful role and Roosevelt had many health problem, he could die earlier, maybe in 1947, shortly after UN's birth.
However it seems as this can not prevent Dewey's victory in 1948 election, that could be bigger due a Roosevelt fatigue after almost twenty years of rule.