While is is true that most DO schools are private, the ones that are public (such as MSUCOM) still need to recoup the taxes that normal in-state people pay from out-of-state applicants. Another reason (this this is a little less intuitive) is that most DO schools are significantly younger than most MD schools. Also, DOs usually go into primary care and make a lot less money over their career than most specialists will. I'll give an example to help illustrate why this is so important. Since Harvard has been around since before America was founded, they have well over 200 years of wealthy alumni giving to their school foundation. A DO school, founded around 1975 or so, pumping out primary care physicians who want to work in underserved (i.e. poor) communities, just simply does not have the money stocked away earning interest for scholarships and grants that an MD school might.
I hope this sheds some light on why DO schools can be a bit more expensive that MD schools.
-Travis-