Can you run wires to an inductive heating coil or will that be a bad idea and why?
That should be fine. The inductive heater works by inducing an eddy current into the metal object that is being heated.

It does so by creating an alternating magnetic field around the coil which by faradays law of induction creates small potentials inside the metal object creating the eddy currents. These small currents cause joule heating of the object which is how the inductive heater warms objects.

The inductance of this coil is very large compared to that of a straight wire in order to produce the required magnetic field. The current flowing through your wires has negligible inductance and will not cause a large enough magnetic field to heat nearby objects.

Think of it like this. Do your TV wires cause near by metal objects to heat up when it draws current?