An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D. Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ∞-sheaves on C and D. An ∞-dense inclusion is dense but the opposite is false in general, see Counterexample 220.127.116.11 in Lurie's Higher Topos Theory.
A nontrivial example of an ∞-dense inclusion of sites is given by the inclusion Cart→Man of cartesian spaces into all smooth manifolds, as explained in Is the site of (smooth) manifolds hypercomplete?
As it turns out, in the 1-categorical case we have an ample supply of dense inclusions of sites, as explained by Proposition C2.2.16 of Johnstone's Sketches of an Elephant: essentially small sites of definition of a given Grothendieck topos E that are subcanonical can be identified with full subcategories of E whose objects form a separating family, with the Grothedieck topology induced from the canonical topology on E.
Every inclusion of such sites is dense, but not necessarily ∞-dense.
I wonder if there is a (practical) criterion that allows us to check whether a dense inclusion is also ∞-dense. We can assume the source site D to be hypercomplete, if necessary.
The example that I have in mind takes D to be the site of smooth manifolds (or cartesian spaces) and C a subcategory of the category of sheaves of sets on smooth manifolds that contains some infinite-dimensional smooth manifolds (e.g., smooth mapping spaces Map(M,N) between finite-dimensional manifolds).
What additional conditions are needed to ensure that a dense inclusion of sites is ∞-dense?