The penny costs more than 3 cents to produce. It is not accepted by most vending machines. It is predominantly stored in jars, left in tip trays, or thrown away. Canada eliminated its penny in 2013. Australia did it in 1992. New Zealand did it in 1990. None of these countries collapsed.
The transition involves rounding cash transactions to the nearest nickel, a rounding that averages out to zero over time and across transactions. No one is systematically made worse off. Several hundred million dollars a year in minting costs disappear.
The penny is a monument to inertia. Inertia is not a good reason to keep a coin that costs more to make than it's worth.