Motivation: Recognizing named entities (NER) and their associated attributes like negation are core tasks in natural language processing. However, manually labeling data for entity tasks is time consuming and expensive, creating barriers to using machine learning in new medical applications. Weakly supervised learning, which automatically builds imperfect training sets from low cost, less accurate labeling rules, offers a potential solution. Medical ontologies are compelling sources for generating labels, however combining multiple ontologies without ground truth data creates challenges due to label noise introduced by conflicting entity definitions. Key questions remain on the extent to which weakly supervised entity classification can be automated using ontologies, or how much additional task-specific rule engineering is required for state-of-the-art performance. Also unclear is how pre-trained language models, such as BioBERT, improve the ability to generalize from imperfectly labeled data. Results: We present Trove, a framework for weakly supervised entity classification using medical ontologies. We report state-of-the-art, weakly supervised performance on two NER benchmark datasets and establish new baselines for two entity classification tasks in clinical text. We perform within an average of 3.5 F1 points (4.2%) of NER classifiers trained with hand-labeled data. Automatically learning label source accuracies to correct for label noise provided an average improvement of 3.9 F1 points. BioBERT provided an average improvement of 0.9 F1 points. We measure the impact of combining large numbers of ontologies and present a case study on rapidly building classifiers for COVID-19 clinical tasks. Our framework demonstrates how a wide range of medical entity classifiers can be quickly constructed using weak supervision and without requiring manually-labeled training data.