Paramagnetic character is due to the presence of unpaired electrons.

1. In $Cr(CO)_6$, Cr is in 0 oxidation state with the electronic configuration $3d^5 4s^1$. CO is a strong field ligand, forcing the pairing of electrons to give a $3d^6$ configuration. All electrons are paired, making it diamagnetic.
2. In $Fe(CO)_5$, Fe is in 0 oxidation state with the electronic configuration $3d^6 4s^2$. Due to the strong field ligand CO, electrons pair up, resulting in a $3d^8$ configuration. All electrons are paired, so it is diamagnetic.
3. In $[Fe(CN)_6]^{4-}$, Fe is in the +2 oxidation state with the electronic configuration $3d^6$. $CN^-$ is a strong field ligand, which pairs up all six electrons in the lower energy $t_{2g}$ orbitals ($t_{2g}^6 e_g^0$). Thus, it is diamagnetic.
4. In $[Cr(NH_3)_6]^{3+}$, Cr is in the +3 oxidation state with the electronic configuration $3d^3$. In an octahedral field, these 3 electrons occupy the $t_{2g}$ orbitals singly ($t_{2g}^3 e_g^0$) according to Hund's rule. Since there are 3 unpaired electrons, the complex exhibits paramagnetic behaviour.