The root of the coming issue has to do with the CMOS battery inside every PS3 and PS4, which the systems use to keep track of the current time (even when they're unplugged). If that battery dies or is removed for any reason, it raises an internal flag in the system's firmware indicating the clock may be out of sync.
After that flag is raised, the system in question has to check in with PSN the next time it needs to confirm the correct time. On the PS3, this online check happens when you play a game downloaded from the PlayStation Store. On the PS4, this also happens when you try to play retail games installed from a disc. This check has to be performed at least once even if the CMOS battery is replaced with a fresh one so the system can reconfirm clock consistency.
On the PS3, the timer check is used to enforce any "time limits" that might have been placed on your digital purchase.
That check seems to be required even for downloads that don't have any set expiration date, adding a de facto one-time online check-in requirement for systems after their internal batteries fail.
The PS4 used the timing check to make sure PSN trophy data is registered accurately and prevent players from pretending to get trophies earlier. Player testing shows this is a requirement to get PS4 games to load.
Last month, Sony shut down PlayStation Store access for the PlayStation 3, PlayStation Vita, and PlayStation Portable. Sony will eventually shut off the PSN servers that power the timing check for the hardware it no longer considers important. After that, it's only a matter of time before failing CMOS batteries slowly reduce all PS3 and PS4 hardware to semi-functional bricks.
Sony could release a firmware update that limits the system functions tied to this timing check, but it hasn't publicly indicated it has any such plans and why should it?