I understand that induction coils can heat up conductive materials such as metals, but I'm wondering if they can heat up normal air. And if they cannot, can ionizing the air beforehand allow it to be heated through induction?
No. Air is (luckily) non conductive otherwise our lives would be very different on this planet. In practice you would not be able to offer up a lump of ionized air to your induction coil, even if you contrived some fabulous magnetic field container the very instant you turned on the induction coil your container would be disrupted by the additional magnetic field of the coil and all the ionized air would decide to go somewhere else (could be messy !).
If you have AC in the coil, it creates alternating magnetic field, which creates alternating electrical field. If the electric field is strong enough, it will cause electrical discharge in the air, the air will become conductive, and the coil will be able to heat air.