Almost all countries have restrictions and limitations to anyone coming into the country legally. However, there are a few that Americans may not visit or go to for any reason. The main one is North Korea due to the danger involved in living in that country. Besides emigrating to these countries, Americans are strongly told to not even visit them. Mainly, these include countries that are at war or conflict with the United States.
There may also be very high rates of crime in these countries. These countries include Angola, Iran, Algeria, Cuba, Syria, Somalia and Afghanistan. Most people who do go to these countries are usually there as soldiers fighting their government or groups of their citizens. They are not there to live or visit.
Each country has their own immigration laws to contend with, and if you are a United States citizen looking to move permanently, you will likely have to give up your American citizenship in addition to learning a decent amount of the local language, proving income, place of residence, and usually fill out a load of paperwork.
Some countries want to be sure you aren't a threat to national security, while others don't want American citizens to become residents at all. These countries usually include some Middle Eastern countries like Iran, as well as North Korea. If you are really intent on becoming a resident of a particular country, you may have to take the long way and denounce U.S. citizenship in favor of another, then denounce that one after you move.