Let $\mathcal{A}$ be a complete and cocomplete Abelian category.
Let $J$ a small category, and let $F_1 \xrightarrow{r} F_2 \xrightarrow{s} F_3$ be a sequence of diagrams/functors $F_i : J\rightarrow \mathcal{A}$ with $r,s$ natural transformations.
Suppose for each object $j\in \mathrm{Obj}(J)$ the sequence $0\rightarrow F_1(j) \xrightarrow {r_j} F_2(j) \xrightarrow {s_j} F_3(j)\rightarrow 0$ is a short exact sequence in $\mathcal{A}$.
If we take the limits (or colimits) $M_i = \mathrm{(co)lim}F_i$ we get an induced sequence $0 \rightarrow M_1 \xrightarrow {\rho} M_2 \xrightarrow{\sigma} M_3 \rightarrow 0$ in $\mathcal{A}$.
In general is the sequence of $M_i$'s necessarily exact?
In complete generality you only preserve one side of the short exact sequence. Taking colimits is a right exact operation while taking limits is a left exact operation and in general neither is two-sided exact. For a counter-example to two-sided exactness, see, for instance, Example 10.8.9 of the Stacks Project. For a different counter-example, try the example on the bottom of page 5 of this reference as well.
By the way, a proof of the fact that taking colimits is always right exact follows from the fact that colimits commute with other colimits and cokernels are colimits in Abelian categories; the statment about limits follows dually.