Equivalence between a derived subcategory and a subcategory of the derived category

Take $C$ to be sheaves of abelian groups on the sphere, and let $A$ be the abelian subcategory of locally constant abelian groups. Then $A$ is equivalent to the category of abelian groups and so $Hom(\mathbb Z,\mathbb Z[2])$ is different depending on whether you take it in $D(A)$ or $D(C)$

This is more comment than answer, but for bounded derived categories we have the following: let $\mathcal{C}$ be an abelian category and $\mathcal{A} \subset \mathcal{C}$ be a Serre subcategory. Then, a sufficient condition for the natural functor $\mathrm{D^b}(\mathcal{A}) \to \mathrm{D}^{\mathrm{b}}_{\mathcal{A}}(\mathcal{C})$ to be an equivalence is that for every exact sequence $S_1:0 \to A \to B \to C \to 0$ in $\mathcal{C}$ with $A \in \mathcal{A}$, there is an exact sequence $S_2: 0 \to A \to B' \to C' \to 0$ in $\mathcal{A}$ and a map of exact sequences $S_1 \to S_2$ which is the identity on $A$.

This result can be found in Keller's ``On the cyclic homology of exact categories'', who in turn refers to SGA5.