I believe a typical key result in number theory whose proof relies (not only, but critically) on functional analysis is the Selberg trace formula and its variants. More generally, the theory of automorphic forms heavily depends on spectral theory, as one can see in this post in Terence Tao's blog.

Besides Hodge and index theories, mentioned in Qiaochu Yuan's comment above as applications of functional analysis to (complex) algebraic geometry and algebraic topology respectively, I believe that a typical key result in number theory whose proof relies (not only, but critically) on functional analysis is the Selberg trace formula and its variants. More generally, the theory of automorphic forms heavily depends on spectral theory, as one can see in this post in Terence Tao's blog.