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The reverse implication seems a little trickier. One way to prove it is to first argue that it depends only on the Quillen equivalence class of $\mathcal{C}$ (in other words, only on the underlying $\infty$-category of $\mathcal{C}$). This lets you assume that $\mathcal{C}$ is a Bousfield localization of simplicial presheaves on some small simplicial category. From here it's not hard to reduce to the case where $\mathcal{C}$ is the category of simplicial sets, in which case the result is true for any uncountable $\kappa$.
One direction is fairly straightforward: if $\mathcal{C}$ is combinatorial, then there exists a cardinal $\kappa$ such that weak equivalences in $\mathcal{C}$ are closed under $\kappa$-filtered colimits. Then $\kappa$-filtered colimits are also homotopy colimits, so the functor $F$ from $\mathcal{C}$ to its underlying $\infty$-category preserves $\kappa$-filtered colimits, and in particular is accessible. It then follows that $F$ preserves $\tau$-compact objects for $\tau$ sufficiently large.
