V1 has no direct effect on the load because no current passes from V1 to the load. It all goes into (or out of) the opamp.
Assuming all resistors being 1k and 0 volt at the noninverting input, then V1 = 1 volt makes the opamp output -1 volt. The current from V1 into the opamp output is 1 - (-1) = 2 volt divided by 1k + 1k = 2k. 2V / 2000 ohm = 1 mA. That current goes from V1, through the resistors, into the opamp output and comes out of the negative supply pin.
This is without any additional load.
If you add a load from the opamp to ground, say a 1 kohm resistor, then, since the output is still -1 volts, the load will force the opamp to drive one more milliamp to the negative supply.
If you add a load from the opamp to ground, say a 1 ohm resistor, then, since the output is still -1 volts, the load will force the opamp to drive one more amp to the negative supply.