My 97 6.5 has an electric pump feeding the lift pump. When the alternator died, I pushed on until the headlights were candles. I don't know if the electric pump was doing anything but the engine was fine. I read running off the lift pump alone can damage it or the IP.

Sixto
87 300D
97 C2500 'burb 6.5

I am not sure why that would happen.
The Lift Pump inside of the Fuel Injection pump in this case provides the pressure and volume needed to push the Plungers out (no springs).

The amount of Fuel Volume pushing the Plungers out determines how far out the Plungers go and the amount that is going to be Injected when the Plungers are pushed in (minus some leakage because the whole IP is lubed by the Diesel Fuel).
This means that if something interferes with the Lift Pump (inside of the IP) Fuel Volume or pressure it is going to effect the amount of Injected Fuel and you are going to notice that because your Engine is not going to run right.
the Lift Pump inside of the IP also provides the pressure to move the Advance Piston inside that advances the Injection Timing.

The only time I think it would be possible to damage the Fuel Injection pump in this case would be if the Lift Pump Inside of the IP was not getting enough Fuel during cranking and some one kept on cranking and cranking.

In the above case there would be no Fuel circulating through the IP to cool or lube it and because the Engine is not running yet you would not notice anything out of the ordinary.

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84 300D, 82 Volvo 244Gl Diesel