Magnetic coupling between two coils for audio amplifier
Will this setup damage my function generator?


If you have a commercial function generator, it's very likely to have a defined output impedance, usually 50ohm, 120ohm or 600ohm depending on the intended use, and be safe into a short circuit. Check the manual, just to make sure. If you have built one yourself, then only you will know. 


  Should I account for impedance matching? 


There is no impedance matching in what you've shown, so nothing to account for. If you want to start getting quantative about what you measure, then the magnetic field produced by the first coil is proportional to the current flowing in it, and the voltage produced by the second is proportional to the rate of change of field in the first. Measure those.


  Should I have the two parts of the circuit connected to a common ground?


As the intended signal coupling is entirely by magnetic field, it doesn't matter. Note that if both your amplifier and function generator plug into mains, you may well have, through the chassis and safety earth. If you connect a second 'earth' connection between them, you may well introduce picked up noise into the amplifier through an unintended loop.