For out-of-warranty work:
TASCAM factory service is usually $60/hr, minimum 1hr. You send it in on your dime, they look at it and give you a quote for repair. If the quote is more than the unit is worth, you can have it sent back at your expense, or abandon it.
My initial guess is that the unit was used with an external battery pack that puts out too much voltage - the DR-680 was not designed for much more than the 12V the official power supply sends out. So it could be just the power supply caps and/or diode that are blown. Those are cheap parts if you do it yourself.