Check the wire going to the fuel shut off solenoid valve for battery voltage with the key switch on, and again with the key switch off.
The fuel solenoid should only have battery voltage when the key is on, and 0 volts when the key is off.
If you find that you have 0 volts when the key is off, but the engine continues to run, the fuel shut off solenoid has stuck in the run position.
I have also run into some head scratcher problems many years past where a defective condition in an electrical system was causing the wiring to the shut off solenoid to stay energized after the key was turned off. But this was on a different piece of machinery, not a forklift.
So, you need to check to see if the shut off wire goes to 0 volts when the key is turned off while the engine is running too.

Fuel shut offs operate to let fuel flow when voltage is applied to them, and to stop the fuel flow when voltage is taken away from them. If it were the other way around, you would never be able to keep the battery from discharging while the engine was shut off.