The proper handling of irregular time series is a significant challenge when formulating predictions from health data. It is difficult to infer the value of any one feature at a given time when observations are sporadic, as a missing feature could take on a large range of values depending on when it was last observed. To characterize this uncertainty directly, we propose a strategy that learns an evidential distribution over irregular time series in continuous time. We demonstrate that this method provides stable, temporally correlated predictions and corresponding uncertainty estimates based on the evidence gained with each collected observation. The continuous time evidential distribution enables flexible inference of the evolution of the partially observed features at any time of interest, while expanding uncertainty temporally for sparse, irregular observations. We envision that this inference process may support robust sequential decision making processes in clinical settings such as feature acquisition or treatment effect estimation.