When and why do hot material take heat from cold material?
The statement is wrong and/or poorly formulated. Heat cannot spontaneously flow from cold areas to hot areas, there are no exceptions known. It contradicts Clausius' statement of the second law and hence also the more general observation/statement that the entropy of an isolated system is increasing.

Of course you can drain heat from cold regions and transport it to hotter regions, but this should always go paired with some other (larger) increase in entropy somewhere else, e.g. by doing work like in a Linde refrigeration. An other example is that some salts have a very high entropy of solvation and can cool water to ~ -15 °C in a lab at room temperature. These salts can be/are used in practical chemistry for cooling a reaction mixture.