The incorrect statement is:

Let us examine each statement one by one, keeping in mind the electronic configurations of the ions that actually decide their magnetic behaviour.

The atomic number of manganese is $$25$$, so in the ground state the electronic configuration of the neutral atom is

$$\text{Mn : }[\,\text{Ar}\,]\,3d^{5}\,4s^{2}.$$

Statement A says that manganate and permanganate ions are tetrahedral. Both ions contain the central manganese atom surrounded by four oxygen atoms and the steric number is $$4$$, giving a tetrahedral geometry. Hence Statement A is correct.

Statement B talks about $$\pi$$-bonding. In these oxyanions, after the $$\sigma$$-framework is formed with $$sp^{3}$$ hybrid orbitals on oxygen, the remaining unhybridised $$p$$-orbitals of oxygen overlap sideways with empty or half-filled $$d$$-orbitals of manganese, giving $$p\!-\!d$$ $$\pi$$-bonding. Thus the statement is in accordance with molecular orbital considerations and is also correct.

Statement C claims that both manganate and permanganate ions are paramagnetic. To verify this we must calculate the number of $$d$$-electrons present in each oxidation state.

1. For the manganate ion, $$\mathrm{MnO_4^{2-}}$$, the oxidation state of manganese is found by writing

$$x + 4(-2) = -2 \;\;\Longrightarrow\;\; x = +6.$$

So manganese is in the $$+6$$ state. We remove six electrons from the neutral atom:

First $$2$$ electrons come out of the $$4s$$ orbital and the next $$4$$ from the $$3d$$ orbitals:

$$[\,\text{Ar}\,]\,3d^{5}\,4s^{2} \xrightarrow{\;-\;2e^-} [\,\text{Ar}\,]\,3d^{5}$$

$$[\,\text{Ar}\,]\,3d^{5} \xrightarrow{\;-\;4e^-} [\,\text{Ar}\,]\,3d^{1}.$$

Thus, in $$\mathrm{MnO_4^{2-}}$$ we have $$3d^{1}$$, i.e. one unpaired electron, so the ion is paramagnetic.

2. For the permanganate ion, $$\mathrm{MnO_4^-}$$, the oxidation state is

$$x + 4(-2) = -1 \;\;\Longrightarrow\;\; x = +7.$$

Now seven electrons are removed from the neutral atom:

$$[\,\text{Ar}\,]\,3d^{5}\,4s^{2} \xrightarrow{\;-\;2e^-} [\,\text{Ar}\,]\,3d^{5}$$

$$[\,\text{Ar}\,]\,3d^{5} \xrightarrow{\;-\;5e^-} [\,\text{Ar}\,]\,3d^{0}.$$

The $$3d$$ subshell is now empty, so no unpaired electrons are present; hence $$\mathrm{MnO_4^-}$$ is diamagnetic.

Therefore, although the manganate ion is paramagnetic, the permanganate ion is diamagnetic. The assertion that both ions are paramagnetic is false.

Statement D saying that the manganate ion is green and the permanganate ion is purple is a well-known experimental fact from qualitative inorganic analysis, so it is correct.

Only Statement C is incorrect.

Hence, the correct answer is Option C.