They may pay a lower percentage of taxes on wealth gains than middle class or upper middle class, due to being able to delay income realization, lower capital gains taxes, and other means that are more accessible to the rich than the middle class or the poor.
But rich people still tend to pay more taxes than less rich people.
Spend 15 minutes talking to an accountant on the premise that you don't want to pay any tax. If you've got enough cash, that's not a problem, anywhere in the World.
The rich pay almost all tax.
"The average American household of 2.64 people receives almost $13,000 worth of federal benefits, services, and protection per annum. These people would have to have a family income of $53,700 to pay as much in taxes as they get in goodies... Only 4.8 percent of the population -- 12,228,000 people -- file income tax returns showing more than $50,000 in adjusted gross income. Ninety-five percent of Americans are on the mooch." -- P. J. O'Rourke [1]
http://taxfoundation.org/article/summary-latest-federal-inco... Shows the top 50% paying nearly all the taxes. That is a far cry from the rich.
So as a simple-language summary we might all agree that the rich pay a majority of the taxes, while the richer half of the country pays nearly all the taxes.
Note that the "richer half" starts at the median income. For the US, the median income is about $52k. That isn't particularly wealthy. When the bar for "nearly all the wealth" is set at that point, it is not surprising at all.