Sadly, there are some printers that somehow read the expiration date out of the cartridge, and even though the cartridge still has a significant amount of ink remaining, will refuse to print with it after it’s"expired". We have an HP 2500C where I work that does that. Complete scam in my opinion.
They will refuse to print after the expiry date (what the OP asked) or after it has supposedly (but not really) run out of ink? How does the printer know what date it is?