Straw Man:
Under the FairTax, everyone gets a check from the government the beginning of each month in order to offset the taxes they'll pay for the bare necessities like food, housing, gasoline, etc. Under the 2011 proposal, the prebate would be $209 per month for an individual adult.
This turns out to be a very progressive system and many leftists have signed on to it. In fact, the FairTax bill in Congress has bipartisan support. Kotlikoff, a left-wing economist who is now challenging Obama in the primaries, has authored studies showing that those on the lower-end of the income ladder would benefit the most from the FairTax. The middle class would benefit moderately, while the wealthy would benefit the least.
It's important to remember that, while low-income and even middle class individuals tend to pay quite low income tax rates, they also pay quite significant payroll taxes both directly and indirectly. And because of the $100k or so cut-off for payroll taxes, the burden of the payroll tax is mainly on middle class Americans. So by completely replacing all currently-existing taxes, the FairTax would significantly cut taxes on the poor and middle class and actually shift the burden a little more onto the wealthy.
Which is quite an amazing accomplishment once you realize that the United States already has the most progressive tax system in the world according to the OECD.