The Gauss-Faraday law, in the covariant form, reads
$ \epsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta,\alpha} = 0, $
while the vacuum field equation is
$ \partial_\mu F^{\mu\nu} = 0. $
When it comes to quantize the electromagnetic field $A^\mu$, only the field equation is considered (as far as I know). So my question is: does the Gauss-Faraday law play any role in QFT?
The Gauss-Faraday law can be viewed as the integrability condition for the existence of the electro-magnetic $4$-potential $A_{\mu}$ via Poincare Lemma.
Or conversely, assuming a globally defined electro-magnetic $4$-potential $A_{\mu}$, the Gauss-Faraday law is trivially satisfied. See also this related Phys.SE post.
So yes, Gauss-Faraday law plays a role already before quantization.
The Gauss-Faraday law in the QFT plays essential role, as well as $\partial_{\mu}F^{\mu\nu} = 0$. In fact, this is related to the fact that in the QFT quantum fields represent irreducible representations of the Poincare group, i.e., the particles. The photon is massless particle with helicities 1, -1, therefore it has only two degrees of freedom, and it can be shown that the corresponding field is $F_{\mu\nu}$ satisfyng above two equations.
