As Zhen Lin said in the comments, there is a very general argument that answers your problem.
Denote $\omega$ for the category of finite ordinals with set-functions between them.
Definition 1. A Lawvere theory is a finite-product-preserving bijective-on-objects functor $\ell \colon \omega^\circ \to \mathcal T$.
A morphism $f$ from the Lawvere theory $\ell_1 \colon \omega^\circ \to \mathcal T_1$ to the Lawvere theory $\ell_2 \colon \omega^\circ \to \mathcal T_2$ is a finite-product-preserving functor $f \colon \mathcal T_1 \to \mathcal T_2$ such that $f \circ \ell_1 = \ell_2$.
Definition 2. A model for the Lawvere theory $\ell \colon \omega^\circ \to \mathcal T$ is a finite-product-preserving functor $\mathcal T \to \mathsf{Set}$. Together with natural transformations, models form a category $\operatorname{Mod}(\ell)$.
Now, if $f$ is a morphism from $\ell_1$ to $\ell_2$, one has a restriction functor from the models of $\ell_2$ to the models of $\ell_1$:
$$ \operatorname{Mod}(\ell_2) \to \operatorname{Mod}(\ell_1), \quad
M \mapsto M\circ f $$
Fact. Let $\mathcal A,B$ be small categories with finite products and $j \colon \mathcal A \to \mathcal B$ a functor between them. If $F \colon \mathcal A \to \mathsf{Set}$ is finite-product-preserving, then so is its left kan extension $j_!F \colon \mathcal B \to \mathsf{Set}$.
Hence, the restriction functor described above admits a left adjoint
$$ \operatorname{Mod}(\ell_1) \to \operatorname{Mod}(\ell_2),\quad
M \mapsto f_!M $$
If $\ell_1$ is the Lawvere theory of heyting algebras and $\ell_2$ the one of boolean algebras, there is a (inclusion) morphism $f \colon \ell_1 \to \ell_2$. The restriction functor is just the 'forgetful' functor you described. It then admits a left adjoint, meaning there is a free functor.
Edit. Also, see this section of the nLab page on Lawvere theories.
