Tensor response regression has been recently introduced to reveal the dependence between the tensor response and the vector predictor. However, existing methods largely overlook the issue of potential heavy tails and outliers. We propose a tensor-t distribution-based framework to tackle this problem. We further discuss the estimation, variable selection and statistical inference through penalized likelihood. An iteratively re-weighting Majorize-Minimization algorithm is developed to efficiently implement our method. Extensive numerical studies support the favorable performance of our proposal in the presence of heavy tails.