One problem here would probably be turbopumps [1] running dry. These things run very close to their limits when fully stressed and would presumably run higher still if they no longer needed to pressurize anything.

The fuel is also used as a coolant [2] so you would not want to run out of fuel in the plumbing before you run out of propellants in the combustion chamber.

[1] https://en.wikipedia.org/wiki/Turbopump

[2] https://en.wikipedia.org/wiki/Merlin_(rocket_engine_family) (search "regeneratively cooled")