The question is essentially the one in the title.
Question. What are some examples of (major) problems in set theory which are solved using techniques outside of mathematical logic?
Inspired by Joel David Hamkin's comment---Simon Thomas has provided applications of various super-ridigity theorems (from the ergodic theory of group actions) to the theory of the Borel complexity of countable equivalence relations, for example he shows that the universal countable equivalence relation is not essentially free
Thomas, Simon, Popa superrigidity and countable Borel equivalence relations, Ann. Pure Appl. Logic 158, No. 3, 175-189 (2009). ZBL1162.03029.
I think the question could in principle have good answers (not turning too much on what is a method from logic) even though no one has found any yet. As a counterfactual hypothetical example, Cantor invented the ordinal numbers motivated by considerations from Fourier analysis. Maybe he also tried using Fourier analysis in his attempts to prove CH. He didn't succeed, but imagine that he did. Fourier analysis is classical mathematics from well outside of logic, so that would be a clearly satisfactory answer. One can imagine more such examples, so the question is really: has anything like that ever actually worked? Maybe not, but could it happen in principle? I dunno. I remember that it's possible to prove the Banach-Tarski paradox using the Hahn-Banach theorem (which is grounded in a weak form of AC), though that probably doesn't "count" since it doesn't actually recast BT as a functional analysis problem.
Dropping down from set theory, in computability theory, there's a famous and surprising theorem of Barrington that the complexity class NC1 can be solved by branching programs of fixed width 5, but not width 4 or less. The reason 5 is the minimum is because S5 (the symmetric group on 5 letters) is not solvable but S4 and smaller are. So that's group theory finding its way into a computability problem. No reason such things can't happen in set theory.
(Too long for a comment, I guess).