Suppose that we have:
1) triangulated categories $C,D$, each equipped with a $t$-structure. 2) triangulated functor $F: C \to D$ which is $t$-exact. 3) $F$ reflects isomorphisms, i.e. is conservative.
Point 3) is equivalent to $F$ reflecting zero objects.
It seems that the restriction of $F$ to the hearts is faithful, as a conservative exact functor between abelian categories.
Is there a nice example when $F$ itself is not faithful?
Here is also a simple example where the restriction of $F$ to the heart is faithful but $F$ itself is not. Let $Vec$ be the abelian category of complex vector spaces and let $Rep(\mathbb{Z})$ be the abelian category of complex representations of the free cyclic group $(\mathbb{Z},+)$. The forgetful functor $U: Rep(\mathbb{Z}) \longrightarrow Vec$ is faithful, conservative and exact. Let $C$ be the derived category of $Rep(\mathbb{Z})$ and $D$ be the derived category of $Vec$, both equipped with their natural $t$-structures. Then $U$ induces a triangulated, $t$-exact, conservative functor $$ F: C \longrightarrow D $$ whose restriction to the corresponding hearts coincides with $U$. However, $F$ is not faithful. To see this, let $T$ be the $1$-dimensional representation of $\mathbb{Z}$ with trivial action and let $V$ be the $2$-dimensional representation of $\mathbb{Z}$ where the generator of $\mathbb{Z}$ acts via the matrix $\left(\begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix}\right)$. There is a non-split short exact sequence in $Rep(\mathbb{Z})$ of the form $$ 0 \longrightarrow T \longrightarrow V \longrightarrow T \longrightarrow 0 $$ classified by a non-trivial map $T \longrightarrow T\left<1\right>$ in $C$. But the corresponding map $F(T) \longrightarrow F(T\left<1\right>) = F(T)\left<1\right>$ is trivial because $Ext^1_{Vec}(\mathbb{C},\mathbb{C}) = 0$.
You can start with the derived category of graded polarizable Hodge structures or with the category of Hodge modules (over a complex variety $X$). These categories possess natural weight structures (along with $t$-structures that are "transversal" to them) and the ("strong") weight complex functor possesses all the properties you want (but it is certainly neiter full nor faithful). Much more detail on these examples can be find in section 2.3 and Remark 2.1.2 of my http://arxiv.org/abs/1011.3507 (that also possess a published version, but the preprint one should be better). Note that the heart of the corresponding $t$-structure for $K^b(Hw)$ is semi-simple!
You may "go even further" and send you category to the direct sum of the "pure" derived categories (so, you consider all $Gr_w$; the target is semi-simple).
