Estimating the causal effect of a treatment on the entire response distribution is an important yet challenging task. For instance, one might be interested in how a pension plan affects not only the average savings among all individuals but also how it affects the entire savings distribution. While sufficiently large randomized studies can be used to estimate such distributional causal effects, they are often either not feasible in practice or involve non-compliance. A well-established class of methods for estimating average causal effects from either observational studies with unmeasured confounding or randomized studies with non-compliance are instrumental variable (IV) methods. In this work, we develop an IV-based approach for identifying and estimating distributional causal effects. We introduce a distributional IV model with corresponding assumptions, which leads to a novel identification result for the interventional cumulative distribution function (CDF) under a binary treatment. We then use this identification to construct a nonparametric estimator, called DIVE, for estimating the interventional CDFs under both treatments based on flexible machine learning methods for conditional distributions. We empirically assess the performance of DIVE in a simulation experiment and illustrate the usefulness of distributional causal effects on two real-data applications.