Given that $x, y, z > 0$, and $x + y + z = xyz$, we have $z(xy - 1) = x + y$. Similarly, $y(zx - 1) = z + x$, and $x(yz - 1) = y + z$. By the AM-GM inequality, we have $xyz = x + y + z \geqslant 3 \sqrt[3]{xyz} \Rightarrow xyz \geqslant 3 \sqrt{3}$. The equality holds if and only if $x = y = z = \sqrt{3}$. Therefore,
$$
\begin{aligned}
& x^{7}(yz - 1) + y^{7}(zx - 1) + z^{7}(xy - 1) \\
= & x^{6}(y + z) + y^{6}(z + x) + z^{6}(x + y) \\
\geqslant & 6 \sqrt[6]{x^{6} y x^{6} z y^{6} x y^{6} x z^{6} x z^{6} y} = 6 \sqrt[6]{x^{14} y^{14} z^{14}} = 6(xyz)^{\frac{7}{3}} \\
\geqslant & 6(\sqrt{3})^{7} = 6 \times 27 \sqrt{3} = 162 \sqrt{3},
\end{aligned}
$$
where the equality holds if and only if $x = y = z = \sqrt{3}$. Hence, the minimum value of the original expression is $162 \sqrt{3}$.

This can be used to solve the following interesting problem:
If $x, y, z \geqslant 0, p \geqslant 2, x + y + z = xyz$, find the minimum value of $x^{p}(yz - 1) + y^{p}(zx - 1) + z^{p}(xy - 1)$.
1 Let positive real numbers $x, y, z$ satisfy $x+y+z=xyz$. Find the minimum value of $x^{7}(yz-1)+y^{7}(zx-1)+z^{7}(xy-1)$.