The mint loses $ on pennies and nickels, but makes $ on all other coins thereby turning a profit of almost $300 million annually.
Nothing.... nothing happens to the economy. If the metal content is worth more than the stated value of the coin, then it becomes a problem for the mint. Because people will smelter the coins and sell the metal, at which point the mint will change the metal content as it has done in the past.
Here in the Great White North we have eliminated the penny. Payments still do come to penny amounts but, when cash is involved, the amount is rounded to the nearest nickel. Please Register or Log in to view the hidden image! Almost there!
The penny may take more than a penny to make but it's largely zinc with a copper coating so it's probably not worth more than a penny if you have to take it apart.
I think owing to technology coins have become obsolete. I collect coins. I like coins. But I almost never use hard currency in commerce. Below is a tool to calculate the melt value of coins. http://www.coinflation.com/coins/basemetal_calc.php The melt value is far less than the nominal value. So there is no problem for the mint.
It seems like false idol worship of gold. Somehow the center of the economy will make up for something that comes out zero in value. This happens all the time, example throwing away food.
I remember when this happened in Italy in the 1970s. All the lowest denomination coins disappeared and you were solemnly given telephone tokens called "gettone" (discs made from some cheap metal with a slot across them), instead - all adding to the sense of barely suppressed chaos that characterised visits to Italy at that time.
The US has done it periodically over the years. The US one cent coin is now copper plated. The first US once cent coins were pure copper. Content has varied over the years. For a brief period during WWII the US produced steel one cent coins owing to metal demands of the war.