Why doesn't the output of the opamp change with the source impedance?
There are other factors to consider such as input bias current (quoted at anything up to 4.5 nA). With 4.5 nA flowing from the input into 100 Mohm, that produces an offset error of anything up to 0.45 volts. That blows the effect of the input resistance out of the water.

But, realistically, with a simulation, it all depends on what is set up in the model parameters. Not all models accurately reflect all op-amp artifacts.


  The datasheet of this opamp says the input impedance is typically
  100Meg


That figure is the differential input impedance and not the input impedance of one input to ground so, it will have much less of an effect. If you look on page 6 of the data sheet you will see that the common mode input impedance is graphed at about 280 Gohm. But like I said, input bias currents will always blow things apart at extremes.