1. Identify the nature of each complex and its usual solubility behavior in water

We compare the given complexes based on whether they are known to be soluble or insoluble in water.

2. Option A: $\mathrm{Fe}_{4}[\mathrm{Fe}(\mathrm{CN})_{6}]_{3}$

This is Prussian blue / ferric ferrocyanide type complex.

• It is a highly polymeric coordination compound.
• Such cyanide-bridged network solids are insoluble in water.

So, A is not soluble.

3. Option B: $(\mathrm{NH}_4)_3[\mathrm{As}(\mathrm{Mo}_3\mathrm{O}_{10})_4]$

This is ammonium arsenomolybdate, a heteropoly compound.

• This compound is classically obtained as a yellow precipitate in qualitative/analytical chemistry.
• Hence it is treated as insoluble or sparingly soluble in water.

So, B is not the soluble one.

4. Option C: $\mathrm{K}_3[\mathrm{Co}(\mathrm{NO}_2)_6]$

This is potassium cobaltinitrite.

• It is used in analysis as a reagent for potassium because the corresponding potassium salt is sparingly soluble / insoluble.
• Hence it is not appreciably soluble in water.

So, C is not soluble.

5. Option D: $[\mathrm{Fe}_3(\mathrm{OH})_2(\mathrm{OAc})_6]\mathrm{Cl}$

This is a basic ferric acetate chloride type trinuclear iron complex.

• Since it is an ionic coordination compound with an outer-sphere chloride ion, it can dissolve in water.
• Basic ferric acetate type complexes are known to be water-soluble.

So, D is soluble in water.

6. Conclusion

The complex that dissolves in water is:

$\boxed{\text{D}}$