Actually the fuel cools the fuel pump. THat is the main reason why you don't want to repeatedly run on empty. It will overheat the pump!
I've seen this come up a few times. Sounds like non sence to me. If the the engine is out of gas, it's not running. If it's not running, the fuel pump is not running. If it's not running, it's not over heating. No gas No heat.
Don't lock this thread down to the pointless neverending land.
Think of the pump as being immersed in a cooling bath of fuel. Remove the bath, it gets hot. Running it right to the end of the fuel capacity means the pump is running in air more than in heat-absorbing fuel. It's an electric motor, after all, and it generates heat, just by having electrons shoved through it.
I don't know if I buy into that myself, but that's what they're sayin'.
Also, in a low-fuel condition there will be moments of sucking air, which unloads the motor, running it up in speed. Kind of like a boat motor when the prop comes out of the water in a good chop. Maybe there's some trouble from that, too.