I really depends on what currency takes over. None of the ones mentioned (Euro, Chinese Renminbi) have the worldwide coverage like the dollar does (which is why it won’t happen).
If it did in some way occur, even then we wouldn’t see that much difference, because the US dollar floats in relation to other currencies and that would still be the case.
@seawulf575
is partially wrong. The reason ZImbabwe and Weimar Germany had hyperinflation is because their economies had nothing that the rest of the world wanted to buy, which made their currencies worth less and less.
As long as the US keeps manufacturing things the rest of the world buys, then the dollar retains its worth to a large degree.