I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of the cohomology presheaf $V\mapsto \mathrm H^\bullet(f^{-1}V,F)$. When $f$ is the identity, the sheafification is zero for $n\geq 1$.
The sheafification of a presheaf $P$ can be constructed by taking mapping $PU$ to the equivalence class of families of sections of $P$ defined on an open cover of $U$, where we identify families which coincide on sufficiently small open covers. A section $s\in PU$ is mapped to the equivalence class it represents.
Hence the fact the above sheafification is zero when $f=1$ expresses the fact sheaf cohomology is global in nature (every cocycle admits a cover on which it's zero).
Using the above construction of sheafification, a section in $\mathbf Rf_\ast F(V)$ is an equivalence class of a family of cocycles $(\Gamma_i\in \mathrm H^\bullet(f^{-1}V_i,F))$ where $(V_i)\twoheadrightarrow V$ is an open cover, and we identify families if they coincide on a sufficiently small preimage of an open cover.
The sheafification map is non-zero for $n\geq 1$ if there's a cocycle of $F$ on some preimage $f^{-1}V$ which is not killed by any preimage of an open cover of $V$.
When $f:X\subset Y$ is a subspace inclusion the above means there's a cocycle of $F$ on $f^{-1}V=X\cap V$ such which does not restrict to zero on any open nighborhood in $f^{-1}V_0=X\cap V_0$ in $X$ of some problematic point $x_0\in X$.
This can't happen for closed embeddings because their pushforward functor is exact.
Question 1. What's an instructive example of a subspace inclusion whose higher direct images are nonzero?
Question 2. For "what kind of maps" $f$ does one expect non-zero higher direct images non-zero? (Examples welcome.)
Lastly, I'd appreciate references with explicit topological examples the derived pushforward.