My experience is on Airbus so I'd be very happy for someone with Boeing fuel system knowledge to correct me on this.
1) Centre tank pumps usually pump at a higher pressure than wing tank pumps so that the centre tank fuel is used up first.
2) When the centre tank is empty sensors in the tank tell the centre tank pumps to stop. By the latter stages of a flight one would expect the centre tank fuel to be used up.
3) It is important that LP tank pumps are immersed in fuel so that they don't introduce air into the system, also the fuel provides a cooling function for the pumps.
4) In some early configurations of the A320 the centre tank pump sensors had a bit of an issue and not switch off the pumps when the tank was empty. This would trigger an ECAM (FUEL CTR TK PUMP LO
PR) and we'd switch off the pumps manually. To avoid damaging the pumps there was an SOP not to switch the centre tank pumps on in the first place unless there was a certain amount of fuel in the tank. Its fixed now - well on our fleet anyway.
5) So, you're probably ahead of me, what if the sensors in the 777 center tank failed or were fooled and the pumps operated whilst the tank was dry? Would EICAS alert on something or could they keep going long enough to put enough air into the system to balls things up?
Centre tank obviously feeds both engines albeit via different LP feeds but it is not impossible for two pumps to be running.
Happy to be shot down - I've only given it a few minutes thought.
Cheers,
Silent Badger