Let $X$ be a smooth variety and $Z \subset X$ be a locally complete intersection (smooth if needed). So $X, Z$ is as good as we need (i am working with toric varieties). Let $\pi : \mathrm{Bl}_Z X \to X$ be the blow up in $Z$ and $p: \tilde{Z} \to Z$ be the exceptional divisor and $j: \tilde Z\to \mathrm{Bl}_Z X$, $ i : Z\to X$.
Is it true that $j_*p^*\mathcal{D}^b(Z) \subset \pi^*\mathcal{D}^b(X)$?
No, that is not true. In fact, these two subcategories are semiorthogonal: for any $F \in D^b(X)$, $G \in D^b(Z)$ one has $$ Hom(j_*p^*G, \pi^*F) = 0. $$ This is a part of Orlov's blowup Theorem (which is stated for smooth $Z$ in smooth $X$, but works as well for lci $Z$ in any $X$).
EDIT. Actually, there are two semiorthogonal decompositions. One looks $$ D^b(Bl_Z(X)) = \langle j_*(p^*D^b(Z) \otimes O(1-c)), \dots, j_*(p^*D^b(Z) \otimes O(-1)), \pi^*D^b(X) \rangle, $$ while the other looks $$ D^b(Bl_Z(X)) = \langle \pi^*D^b(X), j_*(p^*D^b(Z)), \dots, j_*(p^*D^b(Z) \otimes O(c-2)) \rangle. $$ I am using semiorthogonality of the second. In fact, it can be deduced from the semiorthogonality of the first by Serre duality.