We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular, we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a viable experimental technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties.