The Washington Post thinks it has found a fatal flaw in the argument that fast food workers should have higher wages:
"The problem: Fast food is a low-profit margin business. How low? According to Yahoo Finance, 2.4 percent. Just look at the headline: 'Fast-Food Chains Aren’t as Rich as Protesters Think.'"
It is likely that most of the people organizing the push for higher wages in the industry are fully aware of "the problem." If workers got higher wages they would presumably be offset to some extent by lower profits, lower pay for top management, increases in productivity, but there would also be some increase in higher prices.
This would reverse a process whereby fast food prices would have dropped relative to the price of other goods as the wages of workers in the industry fell relative to the economy-wide average. There is no obvious "problem" with this reversal. It essentially means that those on the bottom would enjoy higher real wages and living standards, while those on top would see a relative decline in their living standards. It would only be a relative decline, except for those at the very top, since if the economy is growing normally, higher paid workers could still get a share of productivity gains.