And as the minerals are depleted, the island begins to sink until it has been completely depleted and it settles on the surface, crushing whatever lies below.

If the citizens care enough to ensure it lands in a usable state, and the city has developed a diverse enough economy, it may survive and go on operating as a regular city.

Most would probably be abandoned once depleted, leaving behind large mineshafts and abandoned buildings for nature or squatters to reclaim, which could make for fun dungeons.