This is always true, even without the hypothesis of stability. In an ∞-category a fiber sequence $X\to Y\to Z$ is a pullback square $$\require{AMScd} \begin{CD} X @>>> Y\\ @VVV @VVV \\ * @>>> Z \end{CD}\,.$$

So you are pretty much asking whether the functor $\mathrm{Map}(W,-)$ preserves pullback squares. In fact it preserves all limits. In fact we can write it as the composition of the functors $$ C\to P(C)\xrightarrow{ev_W} \mathcal{S}$$ where the first arrow is the Yoneda embedding and the second is evaluation at $W$.

Proposition 5.1.3.2 of Higher topos theory says that the Yoneda embedding preserves limits, while proposition 5.1.1.2 says that for any functor category (like $P(C)=\mathrm{Fun}(C^{\mathrm{op}},\mathcal{S})$) evaluation preserves (and in fact detects) (co)limits.

This is always true, even without the hypothesis of stability. In an ∞-category a fiber sequence $X\to Y\to Z$ is a pullback square $$\require{AMScd} \begin{CD} X @>>> Y\\ @VVV @VVV \\ * @>>> Z \end{CD}\,.$$

So you are pretty much asking whether the functor $\mathrm{Map}(W,-)$ preserves pullback squares. In fact it preserves all limits. In fact we can write it as the composition of the functors $$ C\to P(C)\xrightarrow{ev_W} \mathcal{S}$$ where the first arrow is the Yoneda embedding and the second is evaluation at $W$.

Proposition 5.1.3.2 of Higher topos theory says that the Yoneda embedding preserves limits, while proposition 5.1.2.3 says that for any functor category (like $P(C)=\mathrm{Fun}(C^{\mathrm{op}},\mathcal{S})$) evaluation preserves (and in fact detects) (co)limits.