Here's an example from commutative algebra. [C = complex numbers]
let S be the ring C[x,y,z] and M be the module C(x,y,z).
Then the projective dimension of M is 2 if the continuum hypothesis holds, and it is 3 if the continuum hypothesis fails.
Drinfeld has pointed out (see http://arxiv.org/abs/math/0309155v4 ) that the set-theoretic problems are coming from using the wrong definition of "projective". Raynaud and Gruson proved that projectivity of a module M is equivalent to the combination of three conditions:
(2) decomposition as a direct sum of countably generated modules
(3) Mittag-Leffler condition
That (2) is possible for projective modules is a theorem of Kaplansky but the decomposition is non-canonical, and this is what introduces the axiom of choice into the proof of "free implies projective".
As I understood it from Drinfeld's lecture, only (1) and (3) are necessary for applications and for developing homological algebra, and (2) is undesirable because it is too strong a condition when working with infinite dimensional bundles. He proposed either "flat and Mittag-Leffler" directly, or a minor variant of that, as a definition of what he called "projectivity with a human face", that would work smoothly in the existing applications and allow a reasonable generalization to the infinite dimensional case.
Also, (1) and (3) are definable in first-order logic, so there is less chance of set theoretic problems from quantification over large or complicated structures.