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How to accurately predict the properties of molecules is an essential problem in AI-driven drug discovery, which generally requires a large amount of annotation for training deep learning models. Annotating molecules, however, is quite costly because it requires lab experiments conducted by experts. To reduce annotation cost, deep Active Learning (AL) methods are developed to select only the most representative and informative data for annotating. However, existing best deep AL methods are mostly developed for a single type of learning task (e.g., single-label classification), and hence may not perform well in molecular property prediction that involves various task types. In this paper, we propose a Task-type-generic active learning framework (termed Tyger) that is able to handle different types of learning tasks in a unified manner. The key is to learn a chemically-meaningful embedding space and perform active selection fully based on the embeddings, instead of relying on task-type-specific heuristics (e.g., class-wise prediction probability) as done in existing works. Specifically, for learning the embedding space, we instantiate a querying module that learns to translate molecule graphs into corresponding SMILES strings. Furthermore, to ensure that samples selected from the space are both representative and informative, we propose to shape the embedding space by two learning objectives, one based on domain knowledge and the other leveraging feedback from the task learner (i.e., model that performs the learning task at hand). We conduct extensive experiments on benchmark datasets of different task types. Experimental results show that Tyger consistently achieves high AL performance on molecular property prediction, outperforming baselines by a large margin. We also perform ablative experiments to verify the effectiveness of each component in Tyger.