What conditions must exist for a polyatomic molecule to be nonpolar?

All atoms in that molecule shall belong to the same element. Otherwise, dipoles must balance each other within the molecule. In such cases, the molecule needs to be symmetric. Explanation

The molecule in question is polyatomic. It contains more than one atoms. That's a rather general definition. Among all non-metals, only the noble gases could form stable monatomic molecules.

Molecules such as N₂, O₂, P₄, and S₈ are nonpolar. All atoms in such molecules belong to the same element. All of their bonds are nonpolar. As a result, the molecules are nonpolar.

Molecules such as CO₂ and CCl₄ are polyatomic. Besides, their atoms belong to more than one element. As a result, some of their bonds are polar. Those polar bonds will create dipoles. However, both CO₂ and CCl₄ are symmetric. Dipoles balance within the molecule. For example, CO₂ is linear. Dipoles from the two polar C=O bonds will line up in opposite directions. They would balance each other, such that the net dipole of the CO₂ molecule is zero. As a result, CO₂ will be a nonpolar polyatomic molecule.

Similarly, C-Cl bonds in CCl₄ are polar. However, CCl₄ molecules are tetrahedral. They are symmetric. Dipoles from C-Cl bonds will add up in all four directions to produce a net dipole of zero. As a result, CCl₄ molecules are nonpolar.