Brouwer and the intuitionist mathematicians denied that a proof by contradiction was valid. Is it demonstrable that certain proofs/theorems then become inaccessible, i.e. without using proof by contradiction it is impossible to show them.
For instance, when I proved that the open cover of a compact set could be reduced to a finite cover, my intuition was that a proof by contradiction was required.
If you mean intuitionistic logic, i.e. logic without LEM, then yes, a lot actually.
For example, Cantor-Bernstein-Schröder theorem, which state that if you have 2 sets $A,B$ with injective from $A$ to $B$ and injective from $B$ to $A$, then there exists bijective from $A$ to $B$.
Apparently CBS is equivalent to LEM under IZF(ZF axioms in intuitionistic logic).
But there are much more! For example, it is consistent that there is injective function from $\Bbb R$ to $\Bbb N$. Or that there is bijection from $2^{2^{\Bbb N}}$ to $\Bbb N$. Or even that all real functions are continuous!