It is given that,

$\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=0$

The figure below represents the triangle $ABC$ and midpoints.

$\mathrm{Ar}\left(△ODE\right)=\frac{1}{2}|\overrightarrow{OD}\times \overrightarrow{OE}|$

$=\frac{1}{2}\left|\left(\frac{\overrightarrow{a}+\overrightarrow{c}}{2}\right)\times \left(\frac{\overrightarrow{b}+\overrightarrow{c}}{2}\right)\right|$

$=\frac{1}{8}\left|(\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{b})\right| ...\left(\mathrm{i}\right)$

The given equation can be written as,

$\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}=\overrightarrow{0} ...\left(\mathrm{ii}\right)$

Multiply $\overrightarrow{b}$ in equation $\left(\mathrm{ii}\right)$ we get,

$\overrightarrow{a}\times \overrightarrow{b}+2(\overrightarrow{b}\times \overrightarrow{b})+3(\overrightarrow{c}\times \overrightarrow{b})=\overrightarrow{0}$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}+3(\overrightarrow{c}\times \overrightarrow{b})=\overrightarrow{0} ...\left(\mathrm{iii}\right)$

Multiplying $\overrightarrow{c}$ in equation $\left(\mathrm{ii}\right)$ we get,

$\overrightarrow{a}\times \overrightarrow{c}+2(\overrightarrow{b}\times \overrightarrow{c})+3(\overrightarrow{c}\times \overrightarrow{c})=\overrightarrow{0}$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{c}+2(\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{0} ...\left(\mathrm{iv}\right)$

Adding equations $\left(\mathrm{iii}\right) \& \left(\mathrm{iv}\right)$ we get,

$\overrightarrow{a}\times \overrightarrow{b}+3(\overrightarrow{c}\times \overrightarrow{b})+\overrightarrow{a}\times \overrightarrow{c}+2(\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{0}$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{b}=\overrightarrow{0}$

Substituting the above value in equation $\left(\mathrm{i}\right)$ we get,

$\mathrm{Ar}\left(△ODE\right)=0$