Does cohomology of coherent sheaves on an complex analytic space commute with coproduct? It is known that on a noetherian scheme $X$ (more generally a noetherian topological space), the canonical morphism of cohomologies of sheaves
$$\bigoplus_{i\in I} H^n(X,F_i)\rightarrow H^n(X,\bigoplus_{i\in I}F_i)$$
are isomrphisms.
My question is whether this morphism still be an isomorphism if $X$ is a complex analytic space (with finite components) and $F_i$ are coherent sheaves on $X$?
 A: At abx's suggestion, here is what used to be a comment now fleshed out as an "answer" (with a mild clarification of a paracompactness hypothesis for the counterexample aspect).
The answer is affirmative in the compact Hausdorff case without any appeal to complex-analytic spaces or coherence conditions.  To be specific, note that since (i) the formation of cohomology of abelian sheaves commutes with finite direct sums and (ii) arbitrary direct sums of both sheaves and of abelian groups are direct limits of finite direct sums (using finite subsets of the index set for the direct sum, the collection of such finite subsets being directed by inclusion), one can recast the question as a special case of the behavior of abelian sheaf cohomology with direct limits:  if $\{F_{\alpha}\}$ is a directed system of abelian sheaves on a topological space $X$ then under what conditions is the natural map
$$\varinjlim {\rm{H}}^n(X, F_{\alpha}) \rightarrow {\rm{H}}^n(X, \varinjlim F_{\alpha})$$
an isomorphism? 
By 4.12.1 in Ch. II of Godement's book "Topologie algebrique et Theorie des Faisceaux" (where the phrase "locally compact" includes a Hausdorff hypothesis, if I remember correctly) one has an analogous such isomorphism property in general for any locally compact and Hausdorff topological space $X$ when the functor ${\rm{H}}^n(X,\cdot)$ is replaced with the functor ${\rm{H}}^n_c(X,\cdot)$ that is the derived functor of the functor $\Gamma_c(X,\cdot)$ of compactly supported sections.  In particular, if $X$ is a compact Hausdorff space then ${\rm{H}}^n(X,\cdot)$ naturally commutes with direct limits of abelian sheaves on $X$ and hence with arbitrary direct sums.

When we drop the compactness condition then within the special context of the question posed (using complex-analytic spaces and coherent sheaves) the answer is always negative beyond the compact case when we impose the very mild topological condition of paracompactness. (It isn't interesting enough to dwell on the non-paracompact case.)  
More precisely, let $X$ be a paracompact (includes Hausdorff) complex-analytic space that is not compact.  Since a Hausdorff topological space is metrizable if and only if it is locally metrizable and paracompact (Smirnov metrization theorem), $X$ is metrizable.  Thus, when $X$ is non-compact there exists a sequence $\{z_n\}$ of pairwise distinct points in $X$ with no convergent subsequence in $X$.  Hence, the subset $Z \subset X$ consisting of these points is closed with the discrete topology as its subspace topology. We regard $Z$ as a discrete analytic space (disjoint union of reduced points), so we have a canonical closed immersion $j:Z \hookrightarrow X$. 
Let $F_n$ be the skyscraper sheaf supported at $z_n$ with stalk $\mathbf{C}$ at $z_n$.
Each $F_n$ is naturally a coherent $O_X$-module and $\oplus F_n = j_{\ast}(O_Z)$ since $Z$ is closed in $X$ with the discrete topology, so
$${\rm{H}}^0(X, \oplus F_n) = {\rm{H}}^0(Z, O_Z) = \prod_n \mathbf{C} = \prod_n {\rm{H}}^0(X, F_n),$$
the second equality using that $Z$ has the discrete topology.  This equality identifies the natural map
$\oplus {\rm{H}}^0(X, F_n) \rightarrow {\rm{H}}^0(X, \oplus F_n)$
with the inclusion $\oplus_{n \ge 1} \mathbf{C} \hookrightarrow \prod_{n \ge 1} \mathbf{C}$ that is not an equality, so the isomorphism question for every non-compact paracompact complex-analytic space $X$ has a negative answer at the level of global sections.
