It's worth delving deeper into the Atlantic article O.J. posted above, and especially a linked article from that one.
The easiest thing to say about this map is that "non-payers" ironically seem more likely to vote Republican and "payers" seem more likely to vote Democratic. But we can't say that for two reasons.
The first reason is that low income earners are much more likely to vote Democratic, even within Republican states. In 2008, Obama lost Georgia by 5 percentage points but he won 70% of voters who earned less than $30,000 -- which is precisely the demo most likely to owe no federal income tax. Obama lost Mississippi by 14 percentage points, but picked up 66% of voters who earned less than $30,000. As a general rule, Republicans win among richer voters -- both in the red states and the blue.
The second factor that complicates our efforts to determine how the 47% vote is that this group is divided between older people and poorer working families. Older people vote in higher numbers. But families earning less than $20,000 voted 30% less than the national average, while households earning more than $150,000 were 30% more likely to vote than average.
Devil's in the details. Basically those "non-income-tax-paying" states simply are the poorest in the country, and the poorest residents (while more likely to vote Democratic) are less likely to vote.
Have to admit, this surprised me a bit. I've generally figured that poor folks in the deep south tend to be ultra-religious, and will vote Republican based on that alone.
 I overlooked this line, which brings this more into focus:
Editor's aside: Voter turnout is also highly correlated with variables like race, but I don't have data on the 47% broken down by those demographics.