Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?
To me, your question seems analogous to saying "I've heard that non-Euclidean geometry requires me to give up Euclid's fifth axiom, which is very useful in many mathematical contexts." You don't have to give up the axiom in the sense of "personally agree, at a basic philosophical level, that it doesn't hold", you just have to think of it in terms like: "As a thought experiment, what sort of mathematics results if we assume it doesn't necessarily hold?"

Also, you might be interested in these blog posts by Andrej Bauer: "The Law of Excluded Middle" and "Proof of negation and proof by contradiction".