So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and $Y$, a continuous map $f:X\rightarrow Y$
and a sheaf of abelian groups $\mathcal{F}$ on $X$ such that the presheaf $f\_\*^{i}(\mathcal{F})$ given by 
$$
U\mapsto H^i(f^{-1}(U),\mathcal{F}|\_{f^{-1}(U)})
$$ 
(for some $i>0$) fails to be a sheaf on $Y$.


I'd like to see the two possible obstructions for $f\_\*^{i}(\mathcal{F})$ to be a sheaf namely


(i) the impossibility of gluing local sections of $f\_\*^{i}(\mathcal{F})$ to a global one,


(ii) the existence of a **non-zero** global section of $f\_\*^{i}(\mathcal{F})$ which is trivial at each stalk $y\in Y$.



Explicit examples presheaves associated to higher direct images which fail to be sheaves
For (i), take $X=S^3$, $Y= S^2$, $f$ the Hopf fibration. There is a local $H^1$ everywhere, coming from the fact that the fibers are circles. This local section is defined consistently everywhere (because there is no monodromy since $Y$ is simply-connected), but it does not glue to a global section because $H^1(X,\mathbb Z)=0$. 


For (ii), simply take $X=Y$ a manifold space with nontrivial cohomology and $\mathcal F$ the constant sheaf. Then there is a global section corresponding to the higher cohomology classes. This section vanishes on every open ball, so it vanishes on every stalk.


