What are some techniques and theorems of analytic number theory that have proved useful outside of number theory?
Look at the list of examples of zetafunctions on Wikipedia, and not all of them are in number theory. Here are some specific applications of the idea of a zetafunction in other areas of mathematics.
I'm not sure if it counts, but Hadamard's factorization theorem in complex analysis was directly inspired by the desire to factor $\zeta(s)$ over its zeros. 


Sieve Theory seems like an example. Originally developped for question related to twinprimes and Golbachlike questions, and other clearly number theoretic questions, such techniques are meanwhile used to address To illustrate this, as one example, let me mention the following result (Corollary 9.7) in a preprint of Kowalski (The principle of the large sieve):
To get a more detailed impression, this preprint or (even better) the book by the same author, The Large Sieve and its Applications (CUP, 2008), into which this preprint developped, would be a good source. 


The Riemannian Zeta function is sometimes important in physics, it occurs both in thermodynamics and in quantum field theory. See for example: http://en.wikipedia.org/wiki/Riemann_zeta_function#Applications 


We can even find applications of analytic number theory techniques outside of mathematics. Think e.g. on the relations between Riemann zeta function and Casimir effect (physics) or Zipf's law (statistics). 


Arithmetic geometry is still arguable number theory I suppose, but this is pretty geometric: in recent years, there have been applications of the Circle Method to the study of rational curves on varieties over finite fields (e.g. http://www.math.columbia.edu/~thaddeus/theses/2011/pugin.pdf). 

