We consider nonlinear functionals of discrete Gaussian free fields with ergodic random conductances on a class of random subgraphs of $\mathbb{Z}^{2}$, including i.i.d. supercritical percolation clusters, where the conductances are possibly unbounded but satisfy an integrability condition. As our main result, we show that, for almost every realisation of the environment, the nonlinear functionals of the rescaled field converge to their continuum counterparts in the Sobolev space $H^{-s}(D)$ for suitable $s > 0$. To obtain the latter, we establish pointwise bounds for the Green's function of the associated random walk among random conductances with Dirichlet boundary conditions, which are valid for all $d \geq 2$.