It's true that voter turnout is inversely correlated to income (or, put another way, that poorer people don't vote as much). We can use data from the U.S. Census Bureau to demonstrate that. In 2012 and 2014, poorer and less educated Americans were indeed less likely to vote, relative to overall turnout.
That's not what we're looking at here, though.
It's not just the case that Sanders lost a lot of states with high income inequality, measured using a metric called the Gini coefficient; it's that his support was actually inversely correlated to income inequality. The less income inequality, the better he did.
That can't be explained as "poor people voting against their self-interest" unless you're also willing to say that wealthy people voted against theirs, too.
As we noted over the weekend, exit polling so far shows that Hillary Clinton has done better with poorer voters in most states. Across all contests, she's got an 11-point advantage with those with a household income of under $50,000 a year. (That's as granular as we can get.)
That's deceptive, too. After all, look at the states where Clinton's margins are much higher: South Carolina, Alabama, Arkansas, Tennessee, Mississippi, etc. Those are also states with larger black populations.
If we consider just the states that voted on March 15, you can see that there's a racial split in preference for the candidates that's larger than the economic one. Clinton won or tied with poorer whites in Florida and Ohio -- but lost with them in Missouri, North Carolina and Illinois. (Missouri didn't have enough non-white voters in the exit poll data to break out by economic range.)
Clinton won all of those states. More turnout from poorer white voters may have helped Sanders win in Missouri, where it was close, but in Ohio and Florida more turnour from poorer white voters wouldn't have helped.
If we go back to the income inequality question, we see the same factor at play. The states with the highest levels of inequality are also largely ones with larger black populations: New York, Louisiana, Florida, Georgia and Mississippi are all in the top six.
The strongest correlation between the results of the Democratic contests so far is between the results and the black population in the state. That also correlates with income inequality, because black Americans are at greater risk of being poor than white ones, for a variety of reasons.
So why has Sanders done worse as income inequality goes up? Because he's done worse with black voters. That's the constant theme of his candidacy, recurring here.