I wonder whether there are problems whose statement do not mention toric varieties (nor simple polytopes, vanishing sets of binomials, etc.), but whose proof nicely and essentially uses them?

To give an idea, two simple random examples -- not for toric varieties, but for other concepts:

• a one-line computation of $H^*(K/T)$, the cohomology of flag variety, using equivariant cohomology, in an answer MSO:21670 by Allen Knutson;
• the hard Lefschetz theorem, that for a Kähler $X$, the map is $\omega^{n-i}: H^i(X) \to H^{2n-i}(X)$ is an isomorphism; the now standard proof by Chern uses representation theory of $sl_2$.