Using exact analytical solutions, it is shown that two recent models of nonlocal electron heat transport are beset with mathematical anomalies leading to unphysical results. First, heat flow with a nonlocal heat flux does not smooth out steep temperature gradients in any finite time. Second, a computation of the thermoelectric field from a vanishing nonlocal current flux is an ill‐posed problem leading to instabilities that violate the very assumption on which the model is based. It is verified that these anomalies can lead to a negative entropy production rate, which implies local thermodynamic instability. In particular, it is proved that a temperature distribution that is initially positive can later become negative for certain nonuniform electron density distributions. These results provide a basic understanding of the various difficulties encountered in numerical implementations of these models.