Assuming that it were possible to do so, if a tunnel from one side of a planet to the other were lined with copper wire, could dropping magnets down the hole be used to generate power?
Not forever. A magnet moving through the coil would generate a current; the current would induce a magnetic field that in turn would slow down the magnet. It's the principle of what is known as 'magnetic braking'. Even if you dug these holes, somehow shielded them from the insane heat and pressure of the nucleus, and vacuum-pumped them so that there's no air resistance, the magnet would fall, lose speed, and arrive at the other end with less energy, so not quite reach the antipodes. It would then keep oscillating, while losing energy, and finally stop in the middle. And all you would have done is harvest the magnet's initial gravitational potential energy.