What is the problem with this circuit? ( MOSFET + OPAMP) Battery protection circuit (draw current from battery)
when the SWITCH is OFF, from V+ (12V) to 1st BAT+ ~30-40mA current
  consumption. Why? What is the problem? In the simulation only ~100uA
  current consumption when the SWITCH is off.


Simulations don't always model every characteristic of a component accurately, particularly when operated outside its normal parameters.    

When the power switch is turned off in your circuit, the voltage on the 'GND' terminal of the LM358 rises towards +12V. However the inverting input of the second op amp is connected directly to +1.2V, so it will become negative relative to GND. 

LM358 inputs are rated for an absolute minimum of -0.3V, but in your circuit it could be getting as much as -10.8V. The LM358 datasheet doesn't say what will happen if you try to pull an input below -0.3V - so I tested it. Current reached 40mA at -1.86V, and increased rapidly as the voltage went more negative. I suspect that in your circuit the LT1236 is dropping about 10V, which is fortunate because that is the only thing stopping the LM358 from blowing up!

To prevent this excessive input current you need to keep the op amp input voltages within their ratings. You could put a 10k resistor in series with the input to limit current, but it will still draw some current when the power is 'off'. A better solution might be to use a voltage divider off battery positive, rather than tapping into the battery at the first cell.