The Cherokee Nation of Oklahoma (CNO) is more concerned about documented traceability and knowing each other than DNA tests. They are (rightly) concerned with the dilution of what it means to be 'Native American.'
This all makes sense when you look at the genetic makeup of the Tribe. Most Cherokees are ethnically 'mixed' - including Bill John Baker, the Chief of the Cherokee Nation, Oklahoma, himself. He's 1/32 Cherokee, or one less generation (potentially) removed than Warren herself.
The Eastern Band Cherokee Nation has no issue with any of this from what they've released so far.
I think there's one more branch today that we haven't heard from on this, but I don't think it matters. I understand where each is coming from on this. What's funny is in 2012, CNO was happy that Warren had said she had Native American genetic history in her family.
Still, it'd be more interesting to actually trace Warren's genetic tree than use a DNA test.