Tensors are widely used to model relationships among objects in a high-dimensional space. The recovery of missing entries in a tensor has been extensively studied, generally under the assumption that entries are missing completely at random (MCAR). However, in most practical settings, observations are missing not at random (MNAR): the probability that a given entry is observed (also called the propensity) may depend on other entries in the tensor or even on the value of the missing entry. In this paper, we study the problem of completing a partially observed tensor with MNAR observations, without prior information about the propensities. To complete the tensor, we assume that both the original tensor and the tensor of propensities have low multilinear rank. The algorithm first estimates the propensities using a convex relaxation and then predicts missing values using a randomized linear algebra approach, reweighting the observed tensor by the inverse propensities. We provide finite-sample error bounds on the resulting complete tensor. Numerical experiments demonstrate the effectiveness of our approach.