Why should field operators satisfy the classical equations of motion?

How to see it in canonical quantization: All operators $\mathcal{O}$ in a quantum theory fulfill the Heisenberg equations of motion $$ \frac{\mathrm{d}}{\mathrm{d}t}\mathcal{O}(t) = \mathrm{i}[H,\mathcal{O}(t)]$$ where $H$ is the Hamiltonian density and which is exactly the quantum version of the classical Hamiltonian equations of motion. So the fields, as they are operators, indeed must obey the classical equations of motion.

How to see it in path integral quantization: Write $\phi'(x) = \phi(x) + \epsilon\delta(y-x)$ and observe the path integral measure is invariant under this. Expand the integrand to first order in $\epsilon$, and deduce $$ \int\left(\frac{\delta S}{\delta\phi(x)} + J(x)\right)\mathrm{e}^{\mathrm{i}S[\phi]+J\phi}\mathcal{D}\phi = 0$$ which is known as the Schwinger-Dyson equation. Setting $J=0$ gives $\delta S/\delta\phi = 0$ (inside the path integral, which is the PI version of "as an operator equation"), which is exactly the classical equation of motion.