In high-stakes domains like medical reasoning, AI assistants powered by large language models (LLMs) are yet to be reliable and safe. We identify a key obstacle towards reliability: existing LLMs are trained to answer any question, even with incomplete context in the prompt or insufficient parametric knowledge. We propose to change this paradigm to develop more careful LLMs that ask follow-up questions to gather necessary and sufficient information and respond reliably. We introduce MEDIQ, a framework to simulate realistic clinical interactions, which incorporates a Patient System and an adaptive Expert System. The Patient may provide incomplete information in the beginning; the Expert refrains from making diagnostic decisions when unconfident, and instead elicits missing details from the Patient via follow-up questions. To evaluate MEDIQ, we convert MedQA and Craft-MD---medical benchmarks for diagnostic question answering---into an interactive setup. We develop a reliable Patient system and prototype several Expert systems, first showing that directly prompting state-of-the-art LLMs to ask questions degrades the quality of medical reasoning, indicating that adapting LLMs to interactive information-seeking settings is nontrivial. We then augment the Expert with a novel abstention module to better estimate model confidence and decide whether to ask more questions, thereby improving diagnostic accuracy by 20.3%; however, performance still lags compared to an (unrealistic in practice) upper bound when full information is given upfront. Further analyses reveal that interactive performance can be improved by filtering irrelevant contexts and reformatting conversations. Overall, our paper introduces a novel problem towards LLM reliability, a novel MEDIQ framework, and highlights important future directions to extend the information-seeking abilities of LLM assistants in critical domains.