Yes, if they replace the display, the Touch ID button has to be replaced together, they must have a way to reprogram it.
From what I know, they will replace the display assembly (the entire front half of the phone) altogether if:
the display is broken,
the top sensors does not work,
the receiver still does not work after replacing just the receiver,
Home button/Touch ID does not work.
The cost for OOW display replacement is around $100, if replacing the display doesn't resolve the issue, they will replace the phone.