That isn't entirely correct. The mint changes coin composition to reduce the intrinsic value of a coin to be less than face value. It is classic debasement. Gold was removed from coins in 1933, silver in 1965, and most of the copper in 1983. In all cases the coin composition was changed because the intrinsic value was greater than the face value.
Paper and ink do have a marginal intrinsic value. It just isn't that much. In areas with rampant counterfitting, the value of a printed bill settles to the market value of the paper and ink.