Background
Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 \rightarrow F(U) \rightarrow G(U) \rightarrow H(U) \rightarrow 0$ is exact for any open set $U$. My solution to this involved the axiom of choice in (what seems to be) an essential way.
Essentially, you are asking to $G(U) \rightarrow H(U)$ to be surjective when you only know that $G \rightarrow H$ is locally surjective. Ordinarily, you might not be able to glue the local preimages of sections in $H(U)$ together into a section of $G(U)$, but since $F$ is flasque, you can extend the difference on overlaps to a global section. This observation deals with gluing finitely many local preimages together. Zorn's lemma enters in to show that you can actually glue things together even if the open cover of $U$ is infinite.
Now, I have not really studied sheaf cohomology, but the idea I have is that it detects the failure of the global sections functor to be right exact. So if you can't even show sheaf cohomology vanishes for flasque sheaves without the axiom of choice, it seems like a lot of the machinery of cohomology would go out the window.
Now, just on the set theoretic level, it seems like there is something interesting going on here. Essentially the axiom of choice is a local-global statement (although I had never thought of it this way before this problem), namely that if $f:X \rightarrow Y$ is a surjection you can find a way to glue the preimages $f^{-1}(\{y\})$ of a surjection together to form a section of the map $f$.
This brings me to my
Questions
Can the above mentioned exercise in Hartshorne be proven without the axiom of choice?
How much homological machinery depends on choice?
Have any reverse mathematicians taken a look at sheaf cohomology as a subject to be "deconstructed"?
Have any constructive set theorists thought about using cohomological technology to talk about the extent to which choice fails in their brand of intuitionistic set theory? (it seems like topos models of such set theories might make the connection to sheaves and their cohomology very strong!)
My google-fu is quite weak, but searches for "reverse mathematics cohomology" didn't seem to bring anything up.
Best Answer
I don't have Hartshorne, so I can't address the specifics of this case. However, there is a very interesting paper by Andreas Blass Cohomology detects failures of the Axiom of Choice (TAMS 279, 1983, 257-269), which addresses questions of this type and should at least put you on the right track.