Only about one-third of the deaths worldwide are assigned a medically-certified cause, and understanding the causes of deaths occurring outside of medical facilities is logistically and financially challenging. Verbal autopsy (VA) is a routinely used tool to collect information on cause of death in such settings. VA is a survey-based method where a structured questionnaire is conducted to family members or caregivers of a recently deceased person, and the collected information is used to infer the cause of death. As VA becomes an increasingly routine tool for cause-of-death data collection, the lengthy questionnaire has become a major challenge to the implementation and scale-up of VA interviews as they are costly and time-consuming to conduct. In this paper, we propose a novel active questionnaire design approach that optimizes the order of the questions dynamically to achieve accurate cause-of-death assignment with the smallest number of questions. We propose a fully Bayesian strategy for adaptive question selection that is compatible with any existing probabilistic cause-of-death assignment methods. We also develop an early stopping criterion that fully accounts for the uncertainty in the model parameters. We also propose a penalized score to account for constraints and preferences of existing question structures. We evaluate the performance of our active designs using both synthetic and real data, demonstrating that the proposed strategy achieves accurate cause-of-death assignment using considerably fewer questions than the traditional static VA survey instruments.