What is the implication for differential privacy if $\epsilon = 0$? In pure differential privacy, the parameter $\epsilon$ represents the desired privacy loss. The smaller the $\epsilon$ is, the more privacy we can obtain. What happens when  we want the privacy loss $\epsilon = 0$. Does that mean we need to keep the data permanently private and we do not answer  any queries from outside ?

Thanks
Yes. 

$\varepsilon$-differential privacy with $\varepsilon=0$ implies that the output of the mechanism must be independent of the input -- i.e., it provides no information whatsoever.