I guess not. However, I think I have to understand proofs of some of them, if not all of them. So what is the criterion, if any? What kind of theorem whose proof I can get away with?
EDIT For example, let me take the main theorems of class field theory. It's rather easy to understand what they say, but it's difficult to understand the proofs. Do I have to understand those proofs to research on algebraic number theory?
EDIT Another example: Hironaka's theorem on resolution of singularities of an algebraic variety in characteristic 0. I guess most algebraic geometers can do their research without understanding its proof.
EDIT I'll add more examples.
The theorem that singular homology groups satisfy the Eilenberg-Steenrod axioms.
Most of the basic theorems of homological algebra, for example, the theorem that a filtered complex has a spectral sequence.
The Feit-Thompson theorem that every finite group of odd order is solvable.
The classification of finite simple groups(CFSG).
EDIT I believe this question is becoming more and more important because mathematic is developing faster and faster than before. I guess you are going to give up understaning the proofs of all the important theorems in your field, regardless whether you want to understand them or not.
EDIT The existence of the field of real numbers. I'm almost certain that one doesn't need to know its proof to do analysis. All one needs to know are its properties.