Where is the finiteness of product used in this proposition from Hartshorne? See this question: Link
I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample.
I have two related questions too:
1) Is an arbitrary product(respectively, coproduct) of quasicoherent sheaves on a scheme quasicoherent?
2) Is a finite product of quasicoherent sheaves on a scheme quasicoherent? 
The products and coproducts are in the category of $\mathcal{O}_X$ modules, NOT that of quasicoherent sheaves. I think I have proven 2), but I'm not sure.
 A: For the questions listed in the body of your post:
In the infinite case: for infinite coproducts over reasonably nice spaces, this is easy. Over a noetherian scheme (or more generally a noetherian topological space) one may check that the direct sum of the local presentations for each quasi-coherent sheaf in the coproduct add together to form local presentations for the direct sum. (Over a non-noetherian topological space, one must take the presheaf given by $U\mapsto \bigoplus_{i\in I}\mathcal{F}_i(U)$ and sheafify.) For products, things are more complicated. Quasi-coherent sheaves on a scheme is a Grothendieck Abelian Category, and in particular has arbitrary products. See here for references to Gabber's proof of this fact.
For the finite case: Yes, and this may be deduced from the fact that finite products and coproducts are equivalent in abelian categories like the category of quasi-coherent sheaves on a scheme (and then one applies the argument involving coproducts from the first paragraph).
To address the issue that comes up in the linked proof, the problem is that the listed exact sequence with direct sums should really be a sequence of direct products, which are not necessarily equivalent to direct sums except when the index set is finite.
