SQL Server 2016 expiration date
SQL Server doesn't stop the service when it hits the expiration. Instead, it is checked when the service starts. If you restart the service, or reboot, or your system crashes, when it tries to come back up, it won't start.

So I'd take your backups, clean SQL Server off the machine entirely (since other components like SSMS might still give you expiration issues) and fix this - before you have to restart the service or reboot and you're not prepared. 

David points out that upgrading from eval to full license is supported, but I avoid that because I’ve had more than one experience where that definitely did not go smoothly. Hopefully this isn’t production, where eval should never have been used in the first place, but in any case I think it is cleaner and safer to nuke and pace than try an in-place upgrade. Those have a variety of potential pitfalls even when an expired eval edition isn’t part of the mix.

I have a post about fully removing evaluation edition, but please don't start this process until you've validated your backups are all good and you've scripted logins, jobs, linked servers etc.


Fun with software : uninstalling SQL Server 2008 R2 Evaluation Edition


The actual date isn't important now, IMHO. If you have sour milk in the fridge, it's sour, and you should throw it away, whether it expired on Easter or on Christmas. Also note that some components may expire sooner, e.g. if you install SSMS along with one instance of eval edition, then install another instance of eval edition later, SSMS will expire with the first one, not the second.