While I can't speak definitively for all aircraft types, the ones I know will experience an engine shutdown if there is a total FADEC failure. The redundancy built into FADECs makes this a rare occurrence, but it can and does happen. That said, far more engine failures are attributed to technical faults within the engine (i.e. blade failures etc) than a FADEC failure. In my outfit, the one and only engine shutdown due to a FADEC failure was actually caused by a bleed air line coupling failure, which pumped raw hot air at the FADEC until it overheated and shut down. Not really the FADEC's fault.