The open mapping theorem is usually proved in most texts using Baires Category theorem which depends upon the axiom of choice.
But if one studies differential calculus in Banach spaces say as in Dieuodenne Foundations of Modern Analysis the theorem is the first part of Inverse mapping theorem( as proved in Walter Rudin's classic Principles of Mathematical Analysis and the proof carries over to Banach space setting ) as a contiuous linear mapis differentiable This proof does not depend upon Baires Category Theorem..
There is a long story of dependence of the Baire's theorem on AC (Axiom of choice). In fact, it comes from the work of P. Bernays in 1942. The Baire's theorem depends on a weaker form of AC, called Axiom of dependent choice (AD).
I am recommending a short Wikipedia article: https://en.wikipedia.org/wiki/Axiom_of_dependent_choice
To study these issues more in-depth, start with the literature at the end of this article. About 99% of the mathematicians who use the Baire's theorem, Banach open mapping theorem, or Banach-Steinhaus theorem, really do not care whether the above depend on AC or DC. Nevertheless, connections with the foundations of mathematics are very important here.
The classic three of functional analysis, Hahn Banach, Open Mapping and Closed Graph theorem can easily be established by Zorn's lemma. Normally the Open Mapping and Closed Graph theory are established by the Baire Category theorem which can be established through Zorn's lemma. However, in these cases the full axiom of choice (equivalent to Zorn's lemma) is overkill and Roman Sznajder states the Baire Category theorem is equivalent to the Axiom of dependent choice. There are constructive proofs of the Hahn Banach theorem in some limited cases of normed spaces but not in it's full generality.
The issue of the foundational axioms all comes back to two concepts, the axiom of choice and the "law of the excluded middle." In many cases many of the theorems of modern mathematics rely on both. Hilbert's basis theorem when first proved (the polynomial of a Noetherian ring is a Noetherian ring), using a non-constructive proof. At that time it was extremely controversial as mathematical Paul Gordan proclaim "this is not mathematics, this is theology." However, Gordon later not only used Hilbert's results but also his methods and admitted they were useful.
Again with care Hilbert's basis theorem can be proved used the axiom of dependent choice, however, it does require the law of the excluded middle. Much of commutative algebra, homological algebra, algebraic geometry, functional analysis, rely on the axiom of choice as of course do results that use these tools.
To some extent - mathematics is a game of logical tautologies. You pick your axioms and let the games begin. Godel pretty much says - there are no complete finite axiom systems (there are some true theorems that will be left on the table in any finite axiom system). There have yet been found internal inconsistencies caused by the axiom of choice. That is probably why 99% of mathematics don't have reservations with using the tools.
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