Offline reinforcement learning (RL) can learn control policies from static datasets but, like standard RL methods, it requires reward annotations for every transition. In many cases, labeling large datasets with rewards may be costly, especially if those rewards must be provided by human labelers, while collecting diverse unlabeled data might be comparatively inexpensive. How can we best leverage such unlabeled data in offline RL? One natural solution is to learn a reward function from the labeled data and use it to label the unlabeled data. In this paper, we find that, perhaps surprisingly, a much simpler method that simply applies zero rewards to unlabeled data leads to effective data sharing both in theory and in practice, without learning any reward model at all. While this approach might seem strange (and incorrect) at first, we provide extensive theoretical and empirical analysis that illustrates how it trades off reward bias, sample complexity and distributional shift, often leading to good results. We characterize conditions under which this simple strategy is effective, and further show that extending it with a simple reweighting approach can further alleviate the bias introduced by using incorrect reward labels. Our empirical evaluation confirms these findings in simulated robotic locomotion, navigation, and manipulation settings.