ltspicespice
It really seems as if Ltspice is indeed flawed, not only did I catch op-amp input terminals taking milliamps of current, but now nmos fets take huge amounts of current into the gate.
Previous post had no mosfets:
LTSpice Has bizzare results for high input impedance devices
Try for yourself.
Circuit:
Gate Current:
Circuit:
Gate Current:
There are different models for the LM358 unit. PSpice simulations based on "LM358" result in a phase margin of app. 50...60 deg. But apparently, this is a very simple model.
However, when using the LM358/NS model the margin is slightly negative! This explains the observed instability during measurements. Hence, external stabilization of the feedback scheme is necessary.
Compensation: A compensation scheme (series connection R=500...1000 Ohms and C=50...100nF) at the opamp output node provides a phase margin of app. 50 deg. (simulation).
As mentioned in comments, opamps don't like their inputs to be as widely different potentials. That said, I believe LT Spice might be telling the truth. Take a look at the LT1002 schematic. Notice the series R and protection diodes between the two inputs. If the two inputs differ by more than 2 diode drops or about 1.2V, the current will be limited by the two 500 series resistors.
The current into the higher potential input and out of the lower potential input will then be equal to the difference between the two Vin sources, minus 1.2V, all divided by 1k ohms (sorry, no math fonts on my phone). 
Best Answer
What you're seeing is perfectly fine - your FET has a huge gate charge and charging that at 400Hz does take some current. The waveform is distorted because dQ/dU of the FET is not constant. (That is, depending on the gate voltage, the capacitance of the gate changes.)
Q_gate = 405nC (from datasheet)
f = 800Hz (gate being charged two times per full cycle)
405nC * 800Hz = 324µA
That's exactly what you're seeing in the simulation.