Does a jet engine keep running when it can no longer be controlled from the cockpit? On the Qantas Flight 32 incident, the Airbus A380's number two engine suffered from an uncontained engine rotor failure and partially damaged the controls of number one engine as well.

According to Wikipedia, engines one and four entered a 'degraded mode' after the explosion, and number one could not be shut off after landing.

So it seems that in case an engine loses all communication signals from the aircraft's control systems in the cockpit, it could somehow remain operational.  To me that definitely sounds a lot safer compared to shutting down and risking the plane lose all thrust.

So do jet engines, by design, keep producing thrust in case they lose contact with the control systems?
General rule here: the engine will remain at its last power setting should the FADECs lose comms with the flight deck controls (in your A380 case) or the thrust lever cables come unhooked (in the case of an older aircraft with mechanical cable-and-pulley controls).

However, the fire handle will kill the engine even if the main controls fail -- it operates the firewall fuel and hydraulic shutoff valves.  In QF32, this didn't happen because the cable for the #1 engine fire handle itself was damaged by shrapnel entering the wing.