It's not so much that you cannot renounce your citizenship, but it is severely discouraged. After some high-profile cases of wealthy individual renouncing citizenship, congress passed laws to make sure that if you want to renounce your citizen ship you need to have paid all of your taxes. Additionally, you may need to pay an additional "exit tax" if you have had significant income or have significant wealth.
As far as becoming another country's citizen, that gets to be fairly complex. Some nations recognize dual citizens, others do not. How that exactly works out depends on the particular circumstances. In any case, the U.S. generally likes to keep its citizens within its grasp.
I guess what was by suggested by viraptor is the absurdity of the situation. What stops Russia to say that any person born on earth is required to fill in tax returns on their entire worldwide income. You enter Russia without paying taxes? Bam -> jail.
They could. And Russia would immediately lose a ton of business. There are no "laws" between countries. Just treaties and agreements. If you don't want to be subject to another countries laws, don't go to that country (or live in a country that enforces other countries laws).
As far as becoming another country's citizen, that gets to be fairly complex. Some nations recognize dual citizens, others do not. How that exactly works out depends on the particular circumstances. In any case, the U.S. generally likes to keep its citizens within its grasp.