What are some examples of "non-logical theorems" proven by Logic?

I was impressed by Bernstein and Robinson's 1966 proof that if some polynomial of an operator on a Hilbert space is compact then the operator has an invariant subspace. This solved a particular instance of invariant subspace problem, one of pure operator theory without any hint of logic.

Bernstein and Robinson used hyperfinite-dimensional Hilbert space, a nonstandard model, and some very metamathematical things like transfer principle and saturation. Halmos was very unhappy with their proof and eliminated non-standard analysis from it the same year. But the fact remains that the proof was originally found through non-trivial application of the model theory.

Another example is the solution to the Hilbert's tenth problem by Matiyasevich. Hilbert asked for a procedure to determine whether a given polynomial Diophantine equation is solvable. This was a number theoretic problem, and he did not expect that such procedure can not exist. Proving non-existence though required developing a branch of mathematical logic now called computability theory (by Gödel, Church, Turing and others) that formalizes the notion of algorithm. Matiyasevich showed that any recursively enumerable set can be the solution set for a Diophantine equation, and since not all recursively enumerable sets are computable there can be no solvability algorithm.

This example is typical of how logic serves all parts of mathematics by saving effort on doomed searches for impossible constructions, proofs or counterexamples. For instance, an analyst might ask if the plane can be decomposed into a union of two sets, one at most countable along every vertical line, and the other along every horizontal line. It seems unlikely and people could spend a lot of time trying to disprove it. In vain, because Sierpinski proved that existence of such a decomposition is equivalent to the continuum hypothesis, and Gödel showed that disproving it is impossible by an elaborate logical construction now called inner model. As is proving it as Cohen showed by an even more elaborate logical construction called forcing.

A more recent example is the proof of the Mordell-Lang conjecture for function fields by Hrushovski (1996). The conjecture "is essentially a finiteness statement on the intersection of a subvariety of a semi-Abelian variety with a subgroup of finite rank". In characteristic $0$, and for Abelian varieties or finitely generated groups the conjecture was proved more traditionally by Raynaud, Faltings and Vojta. They inferred the result for function fields from the one for number fields using a specialization argument, another proof was found by Buium. Abramovich and Voloch proved many cases in characteristic $p$. Hrushovski gave a uniform proof for general semi-Abelian varieties in arbitrary characteristic using "model-theoretic analysis of the kernel of Manin's homomorphism", which involves definable subsets, Morley dimension, $\lambda$-saturated structures, model-theoretic algebraic closure, compactness theorem for first-order theories, etc.

The Tarski–Seidenberg theorem says that the set of semialgebraic sets is closed under projection. It's a pure real-algebraic statement that was originally proved with logic.

Jacobson says this in chapter 5 of his Basic Algebra I:

More generally, Tarski's theorem implies his metamathematical principle that any "elementary" sentence of algebra which is true in one real closed field (e.g., the field of real numbers) is true in every real closed field.

Not sure if this qualifies, but Goodstein's theorem which appears quite number theoretic and states that any Goodstein sequence converges, was proved using ordinals.

So automated theorem provers consist of an application of mathematical logic. Consequently, the solution of the Robbins conjecture by William McCune using EQP qualifies as solving an open problem outside of mathematical logic by using mathematical logic.