If you wish to renounce your citizenship, most countries will allow this only if you hold citizenship in another country. The United States is not one of those countries. In America, it's mainly a matter of paying for the paperwork, which is quite expensive. But then what happens?
You see, should they do this without being a citizen of another nation, they will become what is known as stateless. While difficulties vary for stateless people depending on the country they currently reside in, in general, many otherwise commonplace things can potentially become very problematic for these people- things like getting a job, getting access to education for their kids or themselves, getting citizenship for their potential future children, ability to get married, get a driver’s license, rent or buy a house, travel across borders, or even just getting a bank account. A bigger problem for some is the loss of certain legal protections they’d otherwise be granted.
On this note, in the United States, the over four million stateless people residing in America can potentially be arrested without having committed any crime and jailed for many months, generally while the the U.S. Immigration and Customs Enforcement tries to find a country to deport them to, if possible.
You can see a big can of worms being opened. There are around 12 million stateless people around the world, although most are stateless through no fault of their own. Depending on the country, you can be stateless because your nation collapsed, or you were born in a refugee camp, or even because your parents weren't married. Read about stateless people, by choice or by circumstance, at Today I Found Out.