Can I really take the classical field equations at face value in QFT? To be concrete, let's say I have a relativistic $\phi^4$ theory [with Minkowski signature $(+,-,-,-)$]
$$
\tag{1} \mathcal{L} ~=~ \frac{1}{2} \left ( \partial_{\mu} \phi \partial^{\mu} \phi  - m^2 \phi^2\right ) - \frac{\lambda}{4!} \phi^4.
$$
The classical equation of motion for $\phi$ is:
$$
\tag{2} (\Box + m^2) \phi + \frac{\lambda}{3!} \phi^3 ~=~ 0.
$$
I knew that canonical quantization is basically replacing all Poisson' brackets with (anti-)commutators.  From that point of view, I would expect a classical field equation to remain valid as an operator equation even after quantization.  Am I wrong?
If I am indeed correct, then specifically to the $\phi^4$ example, does that mean
$$
\tag{3} \left \langle \left [ (\Box + m^2) \phi + \frac{\lambda}{3!} \phi^3 \right ] \mathcal{O} \right \rangle ~=~ 0
$$
for any operator $\mathcal{O}$, in the full interacting theory?
And how do I reconcile this with the path integral picture?
Only the classical paths follow classical equations of motion to the letters.  But to quantize a theory, every path is assigned a weight $e^{iS}$, and obviously none of these new inclusions will follow the classical equations.  Then, how can the field equations still hold?
 A: OP's equation is a special case of
$$\left< \Omega \left| T\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~0, \tag{A}$$
where the Euler-Lagrange expression is $\frac{\delta S[\phi;J]}{\delta \phi^{\alpha}(x)}.$
NB: Be aware that the time-ordering usually used in the literature is the covariant time-ordering $T_{\rm cov}$, i.e. time-differentiations inside its argument should be taken after/outside the usual time ordering $T$. This induces quantum corrections/contact terms, so that eq. (A) can be formally rewritten into the standard Schwinger-Dyson equations:
$$ \left< \Omega \left| T_{\rm cov}\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~i\hbar\left< \Omega \left| T_{\rm cov}\left\{\frac{\delta F[\phi]}{\delta \phi(x)} \right\}\right| \Omega \right>_J ~. \tag{B}$$
One should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this & this Phys.SE posts.
