Do you have the Manifold Valve Switches open?

‘707 Captain’ FLIGHT MANUAL Part II – Aircraft and Systems, W01, page 84:

42-45. manifold valve switches
Controls manifold valve.
-When turned horizontally, valve closes.
-When turned vertically, valve opens.

If they are closed when a tank(s) run out of fuel, the fuel will not pass on to other  engines.

AND make sure you turn the Fuel Boost Pump Switches OFF when a tank they are linked to runs out of fuel.

Hope this helps.

Mark