Green wire runs to the Battery light in the dash board.
It won't hurt it to run it that way, but, No, you can't leave it hooked up or battery will go dead when engine is off.
Alternator is basically an electric motor used in reverse, if green wire was left on(12volts) then when engine was off alternator would try to turn fan belt/engine, which of course it can't, but battery would drain as it tried to.
The way the battery light works, it that the alternator is a Ground when it is off.
When you turn on the key one side of the battery light gets battery voltage, the Green wire hooked to the other side of the battery light goes to alternator, which is a Ground now so battery light bulb glows, lights up.
When you start the engine, and alternator is working, then the Green wire now also have 12volts.
With 12volts at both sides of the Light Bulb no power is going thru the light bulb so it is not glowing, it is OFF.
And ff alternator should stop working while engine is running...........battery light comes on.
So it is a pretty simple circuit.
Diagram here:
http://www.therangerstation.com/tech_library/EDiagrams/files/Diagram_charging_1991_2.JPG
You are looking for the LG/R wire, Light Green/Red stripe wire
There is also a resistor with the light bulb, diagram shows it as 510, that is there in case light bulb should burn out, if bulb burned out then no charging like you have now, but with resistor there it would still pass the 12volts to alternator, could be both the bulb and resistor failed but that would be a long shot.
You will have to check under/in dash to see why green wire has no voltage.
If you have a voltage meter in the dash it can run thru there as well