Binary data with one-class missing values are ubiquitous in real-world applications. They can be represented by irregular tensors with varying sizes in one dimension, where value one means presence of a feature while zero means unknown (i.e., either presence or absence of a feature). Learning accurate low-rank approximations from such binary irregular tensors is a challenging task. However, none of the existing models developed for factorizing irregular tensors take the missing values into account, and they assume Gaussian distributions, resulting in a distribution mismatch when applied to binary data. In this paper, we propose Logistic PARAFAC2 (LogPar) by modeling the binary irregular tensor with Bernoulli distribution parameterized by an underlying real-valued tensor. Then we approximate the underlying tensor with a positive-unlabeled learning loss function to account for the missing values. We also incorporate uniqueness and temporal smoothness regularization to enhance the interpretability. Extensive experiments using large-scale real-world datasets show that LogPar outperforms all baselines in both irregular tensor completion and downstream predictive tasks. For the irregular tensor completion, LogPar achieves up to 26% relative improvement compared to the best baseline. Besides, LogPar obtains relative improvement of 13.2% for heart failure prediction and 14% for mortality prediction on average compared to the state-of-the-art PARAFAC2 models.