Let $U^{*}$ denote the pseudocomplement of an element $U$. Then since Heyting algebra homomorphisms preserve pseudocomplements, we shall produce a counterexample by constructing a frame homomorphism which does not preserve pseudocomplements (these frame homomorphisms are quite common).

Let $U\subseteq\mathbb{R}$ be an open set where $U^{**}\neq\mathbb{R}$. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function where $U^{c}=f^{-1}(0)$. Let $\mathcal{T}$ be the collection of all open sets on $\mathbb{R}$. Then $\Omega(f):\mathcal{T}\rightarrow\mathcal{T}$ be the mapping where $\Omega(f)(V)=f^{-1}[V]$ for each open set $V$. Then $\Omega(f)$ is a frame homomorphism. Let $O=\mathbb{R}\setminus\{0\}$. Then $\Omega(f)(O^{**})=\Omega(f)(\mathbb{R})=\mathbb{R}$. However, $\Omega(f)(O)^{**}=U^{**}\neq\mathbb{R}$. Therefore, $\Omega(f(O^{**}))\neq\Omega(f)(O)^{**}$.

It turns out that the complete Heyting homomorphisms between frames are precisely the open mappings between frames. Therefore, every open mapping between topological spaces induces a complete Heyting homomorphism between frames.