This is true if and only if $V$ doesn't occur in the permutation representation of $G$ acting on itself by conjugation (since that's the sum over irreps of $W\otimes W^*$). This is probably the cleanest description you're likely to get.

Of course, one can also state this in terms of characters in which case you want $\sum_{g\in G} \chi_W(g)|C_{G}(g)|=0$.

This is true if and only if $V$ doesn't occur in the permutation representation of $G$ acting on itself by conjugation (since that's the sum over irreps of $W\otimes W^*$). This is probably the cleanest description you're likely to get.

Of course, one can also state this in terms of characters in which case you want $\sum_{g\in G} \chi_v(g)|C_{G}(g)|=0$.