The complementary slackness condition within the KKT optimality conditions states that $\lambda_i^* f_i(x^*) = 0$ for all inequality constraints $f_i(x) \leq 0$ and for optimal primal variable $x^*$ and dual variables $\lambda_i^* \geq 0$. It follows that $\lambda_i^* > 0 \implies f_i(x^*) = 0$ and $f_i(x^*) < 0 \implies \lambda_i^* = 0$.
But in what cases do we get $\lambda_i^* = 0$ and $f_i(x^*) = 0$ simultaneously? This has relevance to applying the implicit function theorem to, e.g., parametric convex optimization problems where it is sometimes assumed that the set $\{i : \lambda_i^* = 0 \text{ and } f_i(x^*, \theta) = 0\} = \emptyset$ (where $\theta$ represents problem parameters). See Assumption 3 in this paper for an example. It isn't immediately clear under what conditions this set would be nonempty.