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.
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:
- If $X$ is hypercomplete and $A=Ind(A_0)$ where $A_0$ has enough projectives (this reduces to the case $A=Ab$).
- 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.
If $A$ is a Grothendieck abelian category then $Sh(X,A)$ is a Grothendieck abelian category, in which case one can endow the category $C(Sh(X,A))$ of unbounded complexes in $Sh(X,A)$ with the injective model structure (weak equivalences are the quasi-isomorphisms and the cofibrations are the monomorphisms) and then take the underlying $\infty$-category $C(Sh(X,A))_{\infty}$. On the other hand, one can take the category $C(A)$ of unbounded chain complexes in $A$, endow it with its own injective model structure, take the associated $\infty$-category $C(A)_\infty$, and then take the $\infty$-category $Sh(X,C(A)_{\infty})$ of $C(A)_{\infty}$-valued $\infty$-sheaves on $X$. There is a natural functor $$ F:C(Sh(X,A))_{\infty} \to Sh(X,C(A)_{\infty}) $$ given by associating to each complex of $A$-sheaves on $X$ the $\infty$-sheafification of the corresponding pre-sheaf of $A$-complexes. This functor is generally not an equivalence, but it is also not too far from being one. To see what's going on, observe that there are two issues here which could make it difficult for $F$ to be an equivalence:
1) The LHS of $F$ is more "rigid" than the RHS. For example, a-priori there is no reason why an $\infty$-sheaf $Open(X)^{op} \to C(A)_{\infty}$ should be realizable as an actual strict sheaf in the discrete category $C(A)$.
2) The LHS and RHS of $F$ have a slightly different notion of equivalence. In particular, equivalences of $\infty$-sheaves of complexes are generally not detected on the level of homology sheaves. This is the stable analogue of the fact that not every $\infty$-topos of the form $Sh(X)$ is hypercomplete.
Issue (1) can be resolved modulu issue (2) by considering the model category $C(PSh(X,A)) = PSh(X,C(A))$ of $C(A)$-valued presheaves on $X$. By a rather general comparison theorem (see Higher Algebra Proposition 1.3.4.25) the underlying $\infty$-category of $PSh(X,C(A))$ is equivalent to the $\infty$-category $PSh(X,C(A)_{\infty})$ of $C(A)_\infty$-valued $\infty$-presheaves on $X$. Using the fact that the ordinary sheafification functor is exact one can show that $C(Sh(X,A))$ is Quillen equivalent to the left Bousfield localization of $C(PSh(X,A))$ with respect to maps of complexes of presheaves which induces an isomorphism on the sheafification of $H_n(-)$ for every $n$. This notion of equivalence is weaker, in general, than the notion of equivalence after $\infty$-sheafification. We may then identify the underlying $\infty$-category $C(Sh(X,A))_{\infty}$ with the localization of $Sh(X,C(A)_{\infty})$ with respect to maps which induce an isomorphism on homology, and identify $F$ with the full inclusion of homology-local objects. In particular, an $\infty$-sheaf can be rigidified to an actual sheaf of complexes if and only if it is homology local.
Note that these homology sheaves can be defined intrinsically. More precisely, the canonical t-structure on $C(A)_{\infty}$ induces a t-structure on $Sh(X,C(A)_{\infty})$, in which the coconnective objects are those which are objectwise coconnective as presheaves. The heart of this t-structure is the category of $A$-valued sheaves on $X$, yielding a natural notion of homology groups which take values in $A$-sheaves. An object is acyclic if it belongs to $Sh(X,C(A)_{\infty})_{\geq n}$ for every $n$. In general, $Sh(X,C(A)_{\infty})$ can have non-trivial acyclic objects (Counterexample 6.5.4.8. in higher topos theory describes an $\infty$-sheaf of chain complexes which is acyclic but not trivial). One may then describe $C(Sh(X,A))_{\infty}$ as the $\infty$-category obtained from $Sh(X,C(A)_{\infty})$ by "dividing out" the acyclic objects.
Finally, all of this also means that $F$ is an equivalence if $Sh(X,C(A)_{\infty})$ has no non-trivial acyclic objects. I didn't check the details, but it seems likely that this phenomenon coincides with the $\infty$-topos $Sh(X)$ being hypercomplete. For example, one should expect $F$ to be an equivalence when $X$ is a paracompact topological space of finite covering dimension (see $\S7$ of higher algebra).