Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e.

$$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$

And we want to know if one also have an equivalence of the $\infty$-topos of sheaves of spaces on these sites:

$$ Sh_{\infty}(\mathcal{C},I) \overset{?}{\simeq} Sh_{\infty}(\mathcal{D},J)$$

This question is about finding an explicit counter example where this is not the case. But I'll give a bit more context.

If my understanding is correct, there are two cases where one can say something:

1) If $ Sh_{\infty}(\mathcal{C},I)$ and $Sh_{\infty}(\mathcal{D},J)$ are hypercomplete, or more generally if one only care about their hypercompletion. Indeed, one always has:

$$ Sh_{\infty}(\mathcal{C},I)^{\wedge} \simeq \ Sh_{\infty}(\mathcal{D},J)^{\wedge}$$

(the $^{\wedge}$ denotes hypercompletion) this is because one can show that they are both equivalent to the $\infty$-category attached to category of simplicial objects of $\mathcal{T}$ with the Jardine-Joyal model structure on simplicial sheaves. And the equivalent of this model structure only depends on the underlying ordinary toposes.

2) If $\mathcal{C}$ and $\mathcal{D}$ have finite limits, then one gets the equivalence:

$$ Sh_{\infty}(\mathcal{C},I) \simeq Sh_{\infty}(\mathcal{D},J)$$

This follows from J.Lurie Lemma 6.4.5.6 in Higher topos theory. This lemma assert that under these assumptions, there is a natural equivalence between

$$ Geom( \mathcal{Y}, Sh_{\infty}(\mathcal{C},I)) \simeq Geom( \tau_0 \mathcal{Y} , Sh(\mathcal{C},I) ) $$

Where $\mathcal{Y}$ is any $\infty$-topos, $Geom$ denotes the spaces of geometric morphsisms, either of $\infty$-topos or ordinary toposes, and $\tau_0 \mathcal{Y}$ is the ordinary topos of homotopy sets of $\mathcal{Y}$. THis in particular gives a universal property to $Sh_{\infty}(\mathcal{C},I)$ which only depends on the $Sh(\mathcal{C},I)$ and so this implies the isomorphisms above.

So what about the general case ? I have quite often heard that this was not true in general, and I'm willing to believe it. But I would really like to see a counter-example !

In some of the places where I have seen asserted that this is not true in general, people were pointing out to the examples where $Sh_{\infty}(C,J)$ is not hypercomplete, and often these examples fall under the assumption of the second case.

In a comment on this old closely related question of mine, David Carchedi mention a counter-example due to Jacob Lurie which indeed avoids both situation... But I havn't been able to understand how it works, and it does not seem to appears in print. If someone can figure it out I'll be very interested to see the details.

Also a counter example to this lemma 6.4.5.6 mentioned above (without the assumption that the site has finite limit) would probably produce an answer immediately. Moreover (assuming such a counterexample exists) there must exists one where the topology is trivial, so I guess there has to be some relatively simple counter examples.