Another example is certain strong forms of Fubini's Theorem.
If you have a real value function on the product of two closed intervals which is bounded, and which is measurable in either coordinate when you fix the other, are the two integrated iterated integrals equal?
(In the actual Fubini's theorem you care about joint measurability, not just measurability on either coordinate.)
If you assume CH, it is easy to construct counterexamples. It turns it is also consistent to have models where it is true.
I don't have a good reference on this, if you know of one please add it in to my answer.
