Let A be an abelian category, and X a topological space.


There are two ways one could try to construct some oo-category of sheaves on X from this data:


1. Consider the category $Sh(X,A)$ of sheaves on X with values in A. This is abelian, so we can consider its derived category; even better, if it has enough injectives, we can enhance this to have an oo-categorical structure for which the derived category is simply the homotopy category. We call $dgSh(X,A)$ the oo-category of complexes of sheaves.
2. Consider the category $Sh(X, dgA)$ of sheaves on X with values in complexes in A. That is, one can make out of A an oo-category dgA whose objects are complexes of A modules. Thus it makes sense to ask for those contravariant (oo-)functors $Open(X)^{op}\to dgA$ which send open covers to Cech objects realizing global sections as homotopy limits.


When are $dgSh(X,A)$ and $Sh(X,dgA)$ equivalent?


I realize I am sweeping issues of boundedness out the picture when constructing dg of an abelian category, for which I apologize; if there are subtleties here, please do tell me. There are also ways to make dgA by favoring projectives, or favoring injectives; I also ignore these issues.



Sheaves of complexes and complexes of sheaves
I'm going to restrict the discussion to Grothendieck abelian categories, because I'm not sure what can be said more generally.
The main reference for what follows is Appendix C in Lurie's book [Spectral Algebraic Geometry](http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf).


The derived ∞-category is a stable ∞-category, but it is the stabilization of a *prestable* ∞-category which is more fundamental. A Grothendieck prestable ∞-category is the connective part of a t-structure on a stable presentable ∞-category where connective objects are closed under filtered colimits. If $C$ is (Grothendieck) prestable, the subcategory $C^\heartsuit$ of discrete objects is (Grothendieck) abelian. 
Prestable ∞-categories are linear analogs of ∞-toposes (insert "Grothendieck" in front of everything):


* topos $\leftrightarrow$ abelian category
* $n$-topos $\leftrightarrow$ abelian $n$-category
* ∞-topos $\leftrightarrow$ prestable ∞-category
* hypercomplete ∞-topos $\leftrightarrow$ separated prestable ∞-category
* Postnikov complete ∞-topos $\leftrightarrow$ complete prestable ∞-category


The procedure for passing from the LHS to the RHS is tensoring with the ∞-category of connective spectra. There are also vertical adjunctions between these: Grothendieck abelian $n$-categories form a coreflective subcategory of Grothendieck prestable ∞-categories, and the separated and complete ones form reflective subcategories of the latter (the same happens on the topos side if you take left exact left adjoint functors as morphisms of ∞-toposes).


According to this picture, given a Grothendieck abelian $A$, there exist three versions of the nonnegative derived ∞-category of $A$:
$$
D^\vee\_{\geq 0}(A) \to D\_{\geq 0}(A) \to D^\wedge\_{\geq 0}(A).
$$
The first is simply the universal one, that is, $A\mapsto D^\vee\_{\geq 0}(A)$ is left adjoint to the functor sending a Grothendieck prestable ∞-category to its heart. The second is the universal *separated* one, and the third is the universal *complete* one. A useful characterization of the middle one is the following: $D\_{\geq 0}(A)$ is the unique separated Grothendieck prestable ∞-category with heart $A$ which is *0-complicial*, meaning that every object admits a $\pi\_0$-epimorphism from a discrete object. 


Stabilizing these prestable ∞-categories, we get stable ∞-categories with t-structures
$$
D^\vee(A) \to D(A) \to D^\wedge(A)
$$
with heart $A$. It turns out the classical derived category of $A$ is the homotopy category of $D(A)$, which explains the notation. The ∞-categories $D^\vee(A)$ and $D(A)$ agree for example if $A$ is compactly generated and every compact object has finite projective dimension. For $A$ the category of abelian groups, all three agree.


If $X$ is a 1-localic ∞-topos and $C$ is a 0-complicial Grothendieck prestable ∞-category, then $Shv(X,C)=X\otimes C$ is also 0-complicial.
In particular, the canonical functor 
$$D^\vee\_{\geq 0} (Shv(X, A)) = D^\vee\_{\geq 0}(X\otimes A) \to X\otimes D^\vee\_{\geq 0}(A) = Shv(X,D^\vee\_{\geq 0}(A))$$
is between $0$-complicial ∞-categories and restricts to an equivalence on the hearts, so it always induces an equivalence between $D\_{\geq 0}(Shv(X,A))$ and the separation of $Shv(X,D^\vee\_{\geq 0}(A))$, which is the same as the separation of $Shv(X,D\_{\geq 0}(A))$.


So **the general answer to your question** is: $D(Shv(X,A))$ is a full subcategory of $Shv(X,D(A))$ and they agree if and only if the t-structure on $Shv(X,D(A))$ is separated. But I don't see an easy way to check that in general. There are two extreme cases that are easy:


1. If $X$ is hypercomplete and $A=Ind(A\_0)$ where $A\_0$ has enough projectives (this reduces to the case $A=Ab$).
2. If $A$ is arbitrary and $X$ has a conservative family of limit-preserving points (this reduces to the case where $X$ is a point).


For example, a concrete sufficient condition for $D(Shv(X,A))=Shv(X,D(A))$ is: $X$ is a paracompact space of finite covering dimension or a Noetherian space of finite Krull dimension, and $A=Ind(A\_0)$ where $A\_0$ is a small abelian category with enough projectives.


**Remark.** If we use $D^\vee$ instead of $D$, it is always true that $Shv(X,D^\vee(A))=D^\vee(Shv(X,A))$ for $X$ a 1-topos. This shows for instance that $Shv(X,D(Ab))$ can be recovered from the abelian category $Shv(X,Ab)$, even if it's not the derived ∞-category.


