We consider the problem of sequential binary hypothesis testing with a distributed sensor network in a non-Gaussian noise environment. To this end, we present a general formulation of the Consensus + Innovations Sequential Probability Ratio Test (SPRT). Furthermore, we introduce two different concepts for robustifying the SPRT and propose four different algorithms, namely the Least-Favorable-Density- SPRT, the Median-SPRT, the M-SPRT, and the Myriad-SPRT. Subsequently, we analyze their suitability for different binary hypothesis tests before verifying and evaluating their performance in a shift-in-mean and a shift-in-variance scenario for different network connectivities and amounts of noise contamination.