Most secure elements used in smart cards are quoted with a minimum useful lifetime of 10 years when stored at a specific temperature. This timeframe doesn't account for EEPROM wear-and-tear, which could reduce the useful lifetime significantly.
It will probably last longer, but no one seems to be able to say how much longer. The sad fact is that EEPROM technology is not designed for long-term data storage. Once the firmware on the chip becomes unreadable, the key will be bricked. Technically the EEPROM is still good, it just forgets what is stored on it (which happens to be critical information).
This is not a problem that is specific to Yubikeys: pretty much every electronic device you own that has microcontrollers with firmware stored in some form of EEPROM will eventually become unusable in a matter of decades, even of all of the other components remain in good operating condition.
