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To justify the existence of measurements that can not be performed jointly on quantum systems, Heisenberg put forward a heuristic argument, involving the famous gamma-ray microscope Gedankenexperiment, based on the existence of measurements that irreversibly alter the physical system on which they act. Today, the impossibility of jointly measuring some physical quantities, termed measurement incompatibility, and irreversible disturbance, namely the existence of operations that irreversibly alter the system on which they act, are understood to be distinct but related features of quantum mechanics. In our work, we formally characterized the relationship between these two properties, showing that measurement incompatibility implies irreversible disturbance, though the converse is false. The counterexamples are two toy theories: Minimal Classical Theory and Minimal Strongly Causal Bilocal Classical Theory. These two are distinct as counterexamples because the latter allows for classical conditioning. Our research followed an operational approach exploiting the framework of Operational Probabilistic Theories. In particular, it required the development of two new classes of operational theories: Minimal Operational Probabilistic Theories and Minimal Strongly Causal Operational Probabilistic Theories. These theories are characterized by a restricted set of dynamics, limited to the minimal set consistent with the set of states. In Minimal Strongly Causal Operational Probabilistic Theories, classical conditioning is also allowed.