Correct Statements Regarding Metal Carbonyls and Synergic Bonding

To determine the correct statements regarding metal carbonyls, let us analyze each statement individually based on the concepts of organometallic bonding and synergic bonding:

1. Statement A: "The metal-carbon bonds in metal carbonyls possess both $\sigma$ and $\pi$-character."

   • In metal carbonyls, bonding consists of a $\sigma$-donation from the carbon atom of $\text{CO}$ to the metal, combined with a $\pi$-backdonation from the metal $d$-orbitals to the $\pi^*$ antibonding orbitals of $\text{CO}$. This dual mechanism imparts both $\sigma$ and $\pi$ character to the metal-carbon ($\text{M-C}$) bond.
   • Therefore, Statement A is correct.

2. Statement B: "Due to synergic bonding interactions between metal and CO ligand, the metal-carbon bond becomes weak."

   • Synergic bonding involves a cooperative effect where $\sigma$-donation increases electron density on the metal, facilitating $\pi$-backdonation. This $\pi$-backdonation imparts partial double-bond character to the $\text{M-C}$ bond, making the metal-carbon bond stronger (and shorter), while weakening the $\text{C-O}$ bond.
   • Therefore, Statement B is incorrect.

3. Statement C: "The metal-carbon $\sigma$ bond is formed by the donation of lone pair of electrons on the carbonyl carbon into a vacant orbital of metal."

   • This describes the primary ligand-to-metal $\sigma$-bond $(\text{L} \rightarrow \text{M})$, formed when the non-bonding electron pair on the carbon atom of the carbonyl group is donated into a vacant orbital of the metal center.
   • Therefore, Statement C is correct.

4. Statement D: "The metal-carbon $\pi$ bond is formed by the donation of electrons from filled d-orbital of metal into vacant $\pi^*$ orbital of CO."

   • This describes the metal-to-ligand $\pi$-backbonding $(\text{M} \rightarrow \text{L})$, where electron density from a filled $d$-orbital of the metal overlaps with the empty antibonding $\pi^*$ orbital of the carbon monoxide molecule.
   • Therefore, Statement D is correct.

Conclusion: Statements A, C, and D are correct.

Hence, the correct option is B.