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Abnormality detection and localization (ADL) have been studied widely in wireless sensor networks (WSNs) literature, where the sensors use electromagnetic waves for communication. Molecular communication (MC) has been introduced as an alternative approach for ADL in particular areas such as healthcare, being able to tackle the shortcomings of conventional WSNs, such as invasiveness, bio-incompatibility, and high energy consumption. In this paper, we introduce a general framework for MC-based ADL, which consists of multiple tiers for sensing the abnormality and communication between different agents, including the sensors, the fusion center (FC), the gateway (GW), and the external node (e.g., a local cloud), and describe each tier and the agents in this framework. We classify and explain different abnormality recognition methods, the functional units of the sensors, and different sensor features. Further, we describe different types of interfaces required for converting the internal and external signals at the FC and GW. Moreover, we present a unified channel model for the sensing and communication links. We categorize the MC-based abnormality detection schemes based on the sensor mobility, cooperative detection, and cooperative sensing/activation. We also classify the localization approaches based on the sensor mobility and propulsion mechanisms and present a general framework for the externally-controllable localization systems. Finally, we present some challenges and future research directions to realize and develop MC-based systems for ADL. The important challenges in the MC-based systems lie in four main directions as implementation, system design, modeling, and methods, which need considerable attention from multidisciplinary perspectives.