Shut down the APU immediately to prevent the fire spreading and burning the tail off? Okay, I don't know the triple at all. That's just an initial reaction!
Edited to add:
Originally Posted by Delta 777QRH
Condition:
Fire is detected in the APU.
APU FIRE switch . . . . . . . . . . . . . . . . . . . Pull, rotate, and hold
Rotate to the stop and hold for 1 second.
If the FIRE APU message remains displayed:
Plan to land at the nearest suitable airport.
Or are you referring to a failure of the APU fire detection system message? In that case, you have to assume your APU fire detection system is inoperative and therefore the APU is being operated without any fire detection capability. If the APU is running, shut it down as soon as you can get by without it. If it's off, don't start it. If it's running and still needed, run it until shutting it down is the safer option. It's a case of weighing the risks between the possibility of an undetected APU fire (remote) and any risks associated with not having the electrical or pneumatic power the APU can provide available. Captain's hat time!
That's just what I gather reading the Delta QRH. Hopefully a qualified 777 pilot will be along shortly to comment.