Here is an article by Kenneth Rogoff on how the U.S. dollar could lose its privileged position relative to other currencies long-term. Basically, this would be bad because suddenly the U.S. would have to pretend as though money is actually real. The government, businesses, and homeowners would have to pay actual interest on their debts, and would have less money to spend on other things. Traveling, working, and living abroad would also get more expensive in dollar terms. On the other hand, imported things would get more expensive but exports would get cheaper for people in other countries to buy, and this might boost trade. There are geopolitical implications too which I don’t understand well.

Anyway, here is how Rogoff says it could go down:

• Up until now, China has pegged its currency to a “basket of currencies” in which the dollar has a fairly large weight. Its currency is lower than it would probably be if traded openly without restrictions. This helps China export cheaply, just as I mentioned above. But eventually they may want to change this, for similar reasons as I mention above.
• Other Asian countries may eventually adopt Chinese currency, peg their currencies to Chinese currency, and start using Chinese currency as reserve savings. This would all increase the status and stability of the Chinese currency internationally relative to the dollar. So far, countries around the world (including China) have kept mostly dollars in reserve because they consider it the most safe, stable currency.