Update
I found the following nice survey paper that lists lots of applications of Hilbert's metric (Birkhoff's version):
Birkhoff's version of Hilbert's metric and its applications in analysis by Bas Lemmens, Roger Nussbaum, April, 2013.
Not Hilbert's metric directly, but immediately related is the Thompson metric which is frequently used.
I list below two interesting examples.
Solving nonlinear matrix equations:
Invariant metrics, contractions and nonlinear matrix equations by H. Lee and Y. Lim
Dynamic Programming:
Thompson metric, contraction property and differentiability of policy functions by L. Montrucchio
