In the unstable case, most fiber sequences are not cofiber sequences or conversely. For instance, if $0\to K\to G\to H \to 0$ is a short exact sequence of groups, then the corresponding maps of classifying spaces$\mathbf{B}K \to \mathbf{B}G \to \mathbf{B}H$ always form a fiber sequence, but not generally a cofiber sequence.

For a concrete counterexample, consider the short exact squence $0 \to \mathbb{Z}\xrightarrow{2} \mathbb{Z}\to \mathbb{Z}/2 \to 0$. Upon taking classifying spaces this becomes $S^1 \to S^1 \to RP^{\infty}$, in which the first map is a double cover whose cofiber is $RP^2$.