Example of a finite Heyting algebra that is not Boolean

The lattice of open sets of any topological space is a Heyting algebra, which is very rarely a Boolean algebra (it is Boolean if and only if every open set is clopen).

The Heyting implication is defined by \begin{align*} U\rightarrow V &= \bigcup\{W\text{ open}\mid U \cap W \subseteq V\}\\ &= (U^c\cup V)^\circ,\end{align*} where $X^c$ is the complement of $X$ and $Y^\circ$ is the interior of $Y$.

Stone's representation theorem for Heyting algebras (HAs) says that every HA is isomorphic to a sub-HA of the HA of open sets of some topological space.

Any finite chain with three or more elements will do.
Check the examples here.

By the way, it also follows from the last paragraph in the section Distributivity in the same article, that every finite distributive lattice is a Heyting algebra.