We consider optimal design of sensor networks for nonlinear Bayesian inverse problems governed by partial differential equations (PDEs). An optimal design is one that optimizes the statistical quality of the solution of the inverse problem. The computed optimal design, however, depends on the modeling assumptions encoded in the governing PDEs or the parameterization of the observation error model. If some of these elements are subject to considerable uncertainties, it is prudent to follow a robust optimal experimental design (ROED) approach. We follow a worst-case scenario approach ROED and develop a scalable computational framework that is suitable for the class of inverse problems under study. Our approach incorporates a probabilistic optimization paradigm for the resulting combinatorial max-min optimization problem. We focus on Bayesian ROED, where the goal is to maximize information gain in presence of uncertainties in the measurement error model. The proposed approach is illustrated in the context of optimal sensor placement for a coefficient inverse problem governed by an elliptic PDE.