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.