How to create this copper coil to withstand much heat
First, as Spehro points out in the comment below, you should make sure your coil matches the inductance of the original cooktop coil.  This will minimize losses in the coil itself, some of which radiates as heat.

Then you should be using copper tubing for your coil, which you run coolant (often water) through.  The currents going through the coil are high frequency anyway, so most of the current is in the outside of the conductor, which is fine for a tube.  You'll have to make sure to properly isolate the coolant from the user, ground, and other metal and circuit parts as it will carry a small current from the coil.

There are a number of diy induction heater examples that use this technique. If you need an example, try a search for them.  The image below shows more than you need.  The transformer and huge capacitor in the upper left are already taken care of by your induction cooktop circuitry.  The interesting part will be the copper tubing, which is connected to the plastic tubing on the lower left which then leads to a pump and a bucket or radiator.