Zener diodes do exhibit current leakage, contrary to the claim that they don't. Every diode, including Zener diodes, has some level of leakage current due to their inherent properties. In reverse bias, while Zener diodes are designed to conduct, they still have a maximum reverse current specified in their datasheets. The discussion highlights a misunderstanding regarding the behavior of Zener diodes in reverse breakdown conditions. Overall, the consensus is that the lecturer's assertion about the absence of leakage current is incorrect.