How is maximum payload determined in the payload-range diagram?
Aircraft also have a Maximum Zero-Fuel Weight (MZFW). This defines the aircraft's maximum weight without any fuel loaded. A large aircraft certainly would not be able to fly with only 10lb of fuel on board, 
so structures like the fuselage are designed to carry a lower payload.

For example, the Boeing 777F freighter has a maximum takeoff weight of 347,814kg. The type certificate lists the MZFW as 248,115kg, which leaves almost 100,000kg for fuel out of the max fuel capacity of 145,538kg. This is a large increase from the passenger version (777-200LR), which has a MZFW of 209,106kg with only slightly lower MTOW.

See also: What are different types of weights of an aircraft?