I'll be using the conventions from Kashiwara & Schapira's Categories and Sheaves book here.
Let $\mathsf{A}$ be an additive category with translation $S: \mathsf{A} \xrightarrow{\sim} \mathsf{A}$. This shift extends to the category of differential objects by changing the sign of the differential. In detail, for a differential object $d_A: A \to SA$, we define its shifted differential object to be $$d_{SA} = - S d_A: SA \to S^2 A.$$ For example, $S=[+1]$ for cochain complexes and $S=[-1]$ for chain complexes. Given a map of differential objects $f: A \to B$, the mapping cone is the object $$\operatorname{Cone}(f) = SA \oplus B$$ equipped with the differential $$d_{\operatorname{Cone}(f)} = \begin{pmatrix} d_{S A} & 0 \\ S f & d_B \end{pmatrix} = \begin{pmatrix} -S d_{A} & 0 \\ S f & d_B \end{pmatrix}. $$ Thus, it follows that $$d_{S\operatorname{Cone}(f)} = -S\begin{pmatrix} d_{S A} & 0 \\ S f & d_B \end{pmatrix} = \begin{pmatrix} S^2 d_{A} & 0 \\ -S^2 f & -S d_B \end{pmatrix}. $$ On the other hand, $$d_{\operatorname{Cone}(S f)} = \begin{pmatrix} d_{S^2 A} & 0 \\ S^2 f & d_{S B} \end{pmatrix} = \begin{pmatrix} S^2 d_{A} & 0 \\ S^2 f & -S d_B \end{pmatrix}. $$ So clearly the Cone construction does not commute with the suspension/shift. Am I doing something wrong here or is this to be expected? If so, why? Does it have something to do with the corresponding stable infinity category?
If it is to be expected, what should the Cocone (homotopy fibre) of $f$ be? Is it $S^{-1}\operatorname{Cone}(f)$ or $\operatorname{Cone}(S^{-1}f)$?
I will first say how the things work in the stable $\infty$-category to get the ''homotopical'' picture clear, and then we will see that even though we can't expect $S(Cf)$ and $C(Sf)$ to be strictly equal, we do get that they are homotopy equivalent (and even isomorphic) as differential objects.
So, in a stable $\infty$-category $\mathscr{C}$, the shift $Sx$ of an object $x\in\mathscr{C}$ is given as pushout $$ \require{AMScd} \begin{CD} x @>>> 0\\ @VVV @VVV\\ 0 @>>> Sx \end{CD} $$ and the cofiber (i.e. mapping cone) $Cf$ of a morphism $f\colon x\to y$ in $\mathscr{C}$ is given as pushout $$ \require{AMScd} \begin{CD} x @>{f}>> y\\ @VVV @VVV\\ 0 @>>> Cf \end{CD} $$ Since $S$ is an equivalence $\mathscr{C}\to\mathscr{C}$, it commutes with pushouts, and therefore we find that the diagram $$ \require{AMScd} \begin{CD} Sx @>{Sf}>> Sy\\ @VVV @VVV\\ 0 @>>> SCf \end{CD} $$ (using $S0\simeq 0$) is also a pushout. But this shows that there is an equivalence $SCf\simeq C(Sf)$ in $\mathscr{C}$ by definition of the right-hand side.
The (homotopy) fiber $Ff$ of $f\colon x\to y$ is given by a pullback square $$ \require{AMScd} \begin{CD} Ff @>>> x\\ @VVV @VV{f}V\\ 0 @>>> y \end{CD} $$ We therefore obtain a rectangle $$ \require{AMScd} \begin{CD} Ff @>>> x @>>> 0\\ @VVV @VV{f}V @VVV\\ 0 @>>> y @>>> Cf \end{CD} $$ of pushout/pullback squares (in a stable $\infty$-category, those are the same things), so the large rectangle is also a pushout/pullback square. That means that $SFf\simeq Cf$, so $Ff\simeq S^{-1}Cf$. However, we also have an equivalence $Ff\simeq C(S^{-1}f)$. To see this, note that after applying $S$ it suffices to produce an equivalence $SFf\simeq SC(S^{-1}f)$. But $$SC(S^{-1}f)\simeq C(SS^{-1}f)\simeq Cf\simeq SFf,$$ so we are done. In particular, we have $S^{-1}Cf\simeq C(S^{-1}f)$ in $\mathscr{C}$.
This is how things work in a stable $\infty$-category $\mathscr{C}$. So, up to homotopy, $S$ commutes with mapping cones. However, because of the homotopical nature of these constructions, there is little hope that they will commute on the nose with each other in a strict model for such a stable $\infty$-category: we only expect a homotopy equivalence between $SCf$ and $C(Sf)$ in your setting. It happens that we can do slightly better (from the perspective of the strict model, not so much from the perspective of the underlying $\infty$-category): we can find an isomorphism between them. Indeed, the map $C(Sf)\to SCf$ given by $$S^2A\oplus SB\xrightarrow{\mathrm{id}\oplus(-\mathrm{id})} S^2A\oplus SB$$ respects the differentials, and is clearly an isomorphism.
This is somewhat special for algebraic examples that often things you expect to be homotopy equivalent are still isomorphic (but also not always, especially if you move on to quasi-isomorphisms as your notion of weak homotopy equivalences). In topological settings you basically never find that constructions that you expect to commute up to homotopy actually commute up to isomorphism in the 1-categorical sense.