The antenna in an NFC-enabled card is a flexible PCB laminated between the plastic outer layers, with the tracks forming a coil round the outside of it. Using some solvent the cards can be easily separated and the antennas retrieved. Once the chip has been removed they can be cleaned up and soldered to, allowing wires to be connected.
What can you do with an NFC antenna? Not a huge amount as you can see in the (Russian language, English subtitles), but he tries it as a not-very-good heating pad, a power harvesting antenna from NFC readers, and perhaps most promisingly, as the coil of a moving-coil speaker. We’re not sure how much effort would be worth making on that last front, but we think with a bit of care there might be room for audible improvement.