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

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.