Is there a name of this problem?
There is no asymptotically better deterministic algorithm than the one that just checks, for each position $i=1,\dots,n$ whether $A[i]=B[i]=C[i]=0$. 

This is easy to see since any correct algorithm must read at least one entry per value of $i$. To see this, suppose that there is an instance
$I = \langle A,B,C \rangle$ for which a correct algorithm $\mathcal{A}$ accesses neither of $A[i]$, $B[i]$, and $C[i]$, for some $i$. 

Consider two instances  $I' = \langle A',B',C' \rangle$  and  $I'' = \langle A'',B'',C'' \rangle$ that are identical to $I$ except (possibly) for the fact that $A'[i]=B'[i]=C'[i]=0$ and $A'[i]=B'[i]=C'[i]\neq 0$. Clearly $\mathcal{A}$  must return the same set $S$ of indices on all of $I$, $I'$, and $I''$. If $i \in S$, then $\mathcal{A}$ fails on $I''$, otherwise $\mathcal{A}$ fails on $I'$.