Can a Playstation 4 be used simultaneously while remote playing with a Vita?

No.

Only one user can use the system at a time.

While multiple users can be logged in, the Vita will display the same as the current view on the TV, and if the Vita user isn't the one with control (you get it from pressing the PS button on the Vita if another user has logged into the console), it can't do anything but watch.

You can't be signed into the same account on both.