The functor $\Omega^\infty\colon Spectra\to Spaces$ which takes a spectrum, replaces it by the associated $\Omega$-spectrum and then takes its $0$th space sends exact triangles to homotopy fiber sequences.
Does $\Omega^\infty$ send exact triangles of spectra to homotopy cofiber sequences of spaces?
For spectra, homotopy cofiber sequences, homotopy fiber sequences and exact triangles are the same.
I think it doesn't, which kinda makes sense since it is a composite of a left and a right adjoint.
More precisely : takes a short exact sequence of abelian groups $0 \to A \to B \to C \to 0$, then you get an exact triangle of spectra $HA \to HB \to HC$. If you rotate, or suspend once you get $\Sigma HA \to \Sigma HB \to \Sigma HC$ which is also exact, but the $0$-spaces are $K(A,1) \to K(B,1) \to K(C,1)$, i.e., the classifying spaces $BA \to BB \to BC$.
Now, the functor $B(-)$ sends short exact sequences of discrete abelian groups to fiber sequences but not cofiber sequences, for example take $0 \to \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \to \mathbb{Z}/2 \to 0$ which becomes $S^1 \to S^1 \to RP^{\infty}$ which is not a cofiber sequence as can be seen in homology.