In metal carbonyls, the bond between the metal and carbon possesses both $\sigma$ and $\pi$ character. The M-C $\pi$ bond is formed by the donation of a pair of electrons from a filled $d$ orbital of the metal into the vacant antibonding $\pi^*$ orbital of carbon monoxide (back-bonding).

1. Effect of Back-Bonding: Filling the antibonding $\pi^*$ orbital of CO weakens the C-O bond, decreasing its bond order and increasing its bond length.
2. Charge on Metal: The extent of back-bonding depends on the electron density on the central metal atom. A higher negative charge on the metal facilitates greater donation of electrons into the $\pi^*$ orbital of CO.
3. Comparison: $[Fe(CO)_4]^{2-}$: Iron has a formal charge of $-2$. (Highest electron density). $[Co(CO)_4]^-$: Cobalt has a formal charge of $-1$. $Ni(CO)_4$: Nickel has a formal charge of $0$. $[Mn(CO)_6]^+$: Manganese has a formal charge of $+1$. (Lowest electron density).
4. Conclusion: Since $[Fe(CO)_4]^{2-}$ has the highest negative charge on the central metal, it exhibits the maximum back-bonding, resulting in the weakest C-O bond and thus the longest C-O bond length .