The debate over the penny's worth (as in its right to exist, not monetary worth, which, of course, is 1/100th of a dollar) is a recurring theme in modern American politics. The first piece of legislation introduced to Congress regarding the matter was the Price Rounding Act of 1989, which was brought forth "to provide a method for removing one-cent pieces from cash transactions." It failed, and Abe Lincoln's shiny head was spared, but the issue is still raised time and time again.
By every measure, minting pennies is a waste of money and resources for the U.S. government. In 2011, the U.S. lost $60.2 million making and circulating the coin and, in 2013, the U.S. Mint estimated that it cost 1.8 cents to produce each penny (not including distribution costs). There are arguments for keeping the coin, but evidence suggests that the penny's existence is pointless.
Were the government to tell the one cent coin to get lost, where would all the pennies go? For answers, it's always best to look to our sane cousins to the north, Canada. Because the two countries' coinage distribution is so similar, the results of a phase-out would likely look the same.
In 2012, Canada's Economic Action Plan started the process of stopping penny production. According to a New York Times article about the phase-out, Canadians were "encouraged to bring [pennies] to banks for eventual melting or to donate them to charities—which will presumably bring them in for melting." Retailers were told to start rounding up or down to the nearest five cent mark starting on February 4, 2013. Still, the government allowed pennies to "be used in cash transactions indefinitely with businesses that choose to accept them."
The U.S. mints a lot of pennies per year—the estimate for 2014's haul is 6,848,400,000—and, were production to cease today, the government would have minted some 300 billion since 1787. Of that, only 140 to 200 billion pennies are actually in circulation today. That's because the coin doesn't have a high usage rate—substantial numbers get thrown into fountains, lost in couch cushions, dropped down subway grates, etc. Should the government enact a phase-out, this will continue to happen to the nation's limited supply of existing pennies (that haven't been voluntarily melted down) until the numbers dwindle enough for collectors to really take notice. How long will this be? Considering you can buy a 5 lb. pile of Canadian pennies for a little over five bucks U.S., chances are it'll take longer than a year.