How can I drive an LC tank circuit to get a nice and clean sine wave?
The way I drove the MOSFET (I used the very same MOSFET, the IRFZ44) was in a linear fashion. The MOSFET was also involved with other transistors and the whole parallel LC (plus transistors) was a highly controlled oscillator. I didn't use a signal generator; whatever tuning capacitance was added to the inductive loop, caused it to operate at a different frequency. This is how thousands per year were made and shipped (and still are).
However, I can't give you the full circuit details but I can advise you on how to drive that MOSFET in a more linear manner: -

Add a source resistor of about 4.7 ohms
Set the DC bias current through the coil/MOSFET to be about 100 mA.
This requires choosing resistors or using a pot
AC couple your input waveform to the MOSFET gate and control its amplitude to get about 22 volts p-p.
Use C0G/NP0 tuning capacitors - yes they are big and expensive but there is no option if you want really decent sinewave purity (and you do need that purity when looking for really small pieces of metal.
The operating frequency for your sized coil should be around 500 kHz to 1 MHz.

Now it should be a reasonable sinewave.