My previous answer left open the following:

**Proposition:** *Let $C$ be a small $\infty$-category with all fiber products, let $\mathcal{T}$ be an $\infty$-topos and let $F : C \rightarrow \mathcal{T}$ be a functor preserving fiber products. Then the left Kan extention:
$$ \widehat{F} : \text{Prsh}(C) \rightarrow \mathcal{T} $$
also preserves fiber product.*

Indeed in the previous answer I showed that the statement was true as soon as $C$ has a terminal object and I have given an example showing that the $1$-categorical version of the proposition is false. But the example I have given somehow suggested that the proposition might be true in the $\infty$-categorical settings. I recently needed an answer to this question so I thought about it, and I think I have a proof of the proposition above.

**Proof:** As $C$ has all fiber product, and $F$ preserves them, one has that for each object $c\in C$, the slice category $C/c$ has all finite limits, and the functor induced by $F$:

$$ F/c : C/c \rightarrow \mathcal{T}/F(c) $$

preserves them. In particular the left Kan extention:

$$ \widehat{F/c} : \text{Prsh}(C/c) \rightarrow \mathcal{T}/F(c) $$

preserve all finite limit by the results of Lurie. This functor is isomorphic to:

$$ \widehat{F}/c : \text{Prsh}(C)/c \rightarrow \mathcal{T}/F(c) $$

It follows in particular that $\widehat{F}$ preserves all fiber product of the form $X \times_c Y$ where $c$ is a representable object. Note that up to this point, everything also works in the $1$-categorical case, and this is exactly how we concluded in the special case where $C$ has a terminal object. The idea is now to use the descent property to extend this to more general pullback.

Let $V \rightarrow X$ be any morphism in $\text{Prsh}(C)$, and write $X$ as the canonical colimits:

$$ X = \underset{c \in Elt(X)}{\text{colim }} c $$

(where $Elt(X)$ is the category of elements of $X$ and I have identified objects of $C$ with their image by the Yoneda embedding)

For each elements $c \rightarrow X$, let $V_c$ be the fiber of $V \rightarrow X$ over $c$.

By descent, one has:

$$ V = \underset{c \in Elt(X)}{\text{colim }} V_c $$

Now, as $\widehat{F}$ preserves all pullbacks whose bottom corner is representable, it follows that the the natural transforation: $\widehat{F}(V_c) \rightarrow F(c)$ is cartesian (in $c$), and hence, by descent in $\mathcal{T}$ it follows that all the maps $\widehat{F}(V_c) \rightarrow F(c)$ are pullback of the maps between the colimits:

$$ \widehat{F}(V) \simeq \underset{c \in Elt(X)}{\text{colim }} \widehat{F}(V_c) \rightarrow \underset{c \in Elt(X)}{\text{colim }} F(c) \simeq \widehat{F}(X)$$

Which shows that $\widehat{F}$ also preserves all pullback of the form $V \times_X c$ as soon as $c$ is representable.

To conclude, one either use a again a similar (but simpler) colimit/descent argument in the variable $c$, or one simply write (using descent) that the functor $\text{Prsh}(C)/X \rightarrow \mathcal{T}/\widehat{F}(X)$ can be decomposed as a limits:

$$ \text{Prsh}(C)/X \simeq \lim_{c \in Elt(X)} \text{Prsh}(C)/c \rightarrow \lim_{c \in Elt(X)} \mathcal{T}/F(c) \simeq \mathcal{T}/\widehat{F}(X) $$

(Note: one needs the fiber product preservation proved above to show this)

But as all the functors $\text{Prsh}(C)/c \rightarrow \mathcal{T}/F(c)$ preserves finite limits, their limits also preserve finite limits and hence for all $X \in \text{Prsh}(C)$ the functor $\text{Prsh}(C)/X \rightarrow \mathcal{T}/\widehat{F}(X)$ preserve all finite limits, which concludes the proof.