I just found an example, so I thought it would be good to post it here, but if anyone knows other examples, or a more general way to construct some I would be interested to see them as well.

This example comes relatively immediately from the example of $1$-site producing a non Hypercomplete $\infty$-topos in the paper of Dugger, Hollander and Isaksen Hypercovers and simplicial presheaves.

They are looking at the site whose underlying category is the posets $I$:

$$ \require{AMScd} \begin{CD} V_0 @<<< U^l_0 \\ @AAA @AAA \\ U^r_0 @<<< V_1 @<<< U_1^l \\ @. @AAA @AAA \\ @. U^r_1 @<<< V_2 @<<< U^l_2 \\ @. @. @AAA @AAA \\ @. @. \vdots @<<< \vdots @<<< \dots \end{CD} $$

And where for each $i$, $U_i^r, U_i^l$ forms a cover of $V_i$. It is not too hard to check that this defines a topology. They show that the $\infty$-topos of sheaves of spaces on this site is not Hypercomplete.

But if one now look at the category of sheaves of sets. Then One can apply the comparison lemma. Let $J \subset I$ be the full subcategory on the $U^{l/r}_i$, then as each $V_i$ is covered by the $U^{l/r}_i$ so by the comparison lemma, the category Sh(I) and Sh(J) (for the induced topology) are equivalent. But the topology induced on $J$ is trivial so one has:

$$Prsh(J) \simeq Sh(I) $$

One the other hand $Prsh_{\infty}(J)$ is obviously hyperconnected while $Sh_{\infty} (I)$ has been proved to not be hyperconnected in the paper mentioned above, so:

$$ Prsh_{\infty}(J) \not\simeq Sh_{\infty} (I) $$

Also note that this example of Dugger-Hollander-Isaksen is closely related to J.Lurie's example involving the Hilbert cube. Indeed if $Q= [0,1]^{\mathbb{N}}$ is the Hilbert Cube, then defining:

$$ U_i^l = ]0,1[^i \times [0,1[ \times Q $$ $$ U_i^r = ]0,1[^i \times ]0,1] \times Q $$ $$ V^i = U_{i-1}^r \wedge U_{i-1}^l = ]0,1[^{i} \times Q $$

Gives a full subcategory of $\mathcal{O}(Q)$ (the category of opens of $Q$) stable under intersection and isomorphic to the $I$ above, with the induced topology on $I$ being the topology described by Dugger-Hollander-Isaksen. So one has a geometric morphisms from $Sh(Q)$ to $Sh(I)$. This somehow show how something along the line of the comment of David Carchedi linked in the question might produces an examples (though as I understand it one might needs to consider more open subset than what the comment said, but I might still be missing something)

I just found an example, so I thought it would be good to post it here, but if anyone knows other examples, or a more general way to construct some I would be interested to see them as well.

This example comes relatively immediately from the example of $1$-site producing a non Hypercomplete $\infty$-topos in the paper of Dugger, Hollander and Isaksen Hypercovers and simplicial presheaves.

They are looking at the site whose underlying category is the posets $I$:

$$ \require{AMScd} \begin{CD} V_0 @<<< U^l_0 \\ @AAA @AAA \\ U^r_0 @<<< V_1 @<<< U_1^l \\ @. @AAA @AAA \\ @. U^r_1 @<<< V_2 @<<< U^l_2 \\ @. @. @AAA @AAA \\ @. @. \vdots @<<< \vdots @<<< \dots \end{CD} $$

And where for each $i$, $U_i^r, U_i^l$ forms a cover of $V_i$. It is not too hard to check that this defines a topology. They show that the $\infty$-topos of sheaves of spaces on this site is not Hypercomplete.

But if one now look at the category of sheaves of sets. Then One can apply the comparison lemma. Let $J \subset I$ be the full subcategory on the $U^{l/r}_i$, then as each $V_i$ is covered by the $U^{l/r}_i$ so by the comparison lemma, the category $Sh(I)$ and $Sh(J)$ (for the induced topology) are equivalent. But the topology induced on $J$ is trivial so one has:

$$Prsh(J) \simeq Sh(I) $$

One the other hand $Prsh_{\infty}(J)$ is obviously hyperconnected while $Sh_{\infty} (I)$ has been proved to not be hyperconnected in the paper mentioned above, so:

$$ Prsh_{\infty}(J) \not\simeq Sh_{\infty} (I) $$

Also note that this example of Dugger-Hollander-Isaksen is closely related to J.Lurie's example involving the Hilbert cube. Indeed if $Q= [0,1]^{\mathbb{N}}$ is the Hilbert Cube, then defining:

$$ U_i^l = ]0,1[^i \times [0,1[ \times Q $$ $$ U_i^r = ]0,1[^i \times ]0,1] \times Q $$ $$ V^i = U_{i-1}^r \wedge U_{i-1}^l = ]0,1[^{i} \times Q $$

Gives a full subcategory of $\mathcal{O}(Q)$ (the category of opens of $Q$) stable under intersection and isomorphic to the $I$ above, with the induced topology on $I$ being the topology described by Dugger-Hollander-Isaksen. So one has a geometric morphisms from $Sh(Q)$ to $Sh(I)$. This somehow show how something along the line of the comment of David Carchedi linked in the question might produces an examples (though as I understand it one might needs to consider more open subset than what the comment said, but I might still be missing something)