Horrendous disorientation has happened before, but that would require both pilots to have been extremely inattentive to flight time amongst other things.
Also, the plane wouldn't just turn off on reaching its destination, its not a cheap satnav!
Well it is - over ADS-B, unless the transponder is turned off, and that's also of no use if its not in coverage for an ADS-B receiver. (not posting to contradict here). The amount of people who assume that GPS is two-way still astounds me.