California archaeologist here, currently supervising the analysis of human remains from several archaeological sites in my region. There are several ways to answer this question. Ancient DNA analysis and radiocarbon dating of human remains does happen regularly, but not often. It depends greatly on which tribe is involved and what their preferences are.

I'm an archaeologist, not a Native person, so please bear in mind that this is my personal view and I'm not trying to speak for anyone else here. My sense is that many people in the Native community are often uncomfortable with destructive analysis of human remains. This is sometimes for spiritual reasons, but also has a lot to do with the fact that there has historically been an incredible amount of violence toward Native American people and their bodies - which in the past also extended to the treatment of human remains, which were excavated and often treated carelessly, or at least without great respect, by archaeologists. (The being treated as a "subject" that you mention.) Given this, destroying bones for DNA or other analysis evokes feelings of trauma for some (but not all) Native people.

There is also the issue that DNA analysis may tell a story which contradicts tribal views of their own origins, if the individuals analysed have a significantly different DNA than contemporary tribal people in the area. While archaeologists have identified movements of peoples and cultures in prehistory based on artifact types (and human remains), many tribal people hold the belief that their ancestors were present in their homelands from the creation of the world onward. This leads to hesitancy, especially because there's a long history in North America of Euro-Americans trying to appropriate the ancient history of the continent and erase Native American cultures and accomplishments. Witness all the books "proving" that the ancient people of the Americas were Europeans, Israelites, Phoenicians, Africans, Polynesians, Vikings, or some long-lost 'Caucasian' immigration from the European Paleolithic. Euro-American culture has had a hard time overcoming a long tradition of racism and accepting the reality of Native American cultures and civilizations. Given that, DNA analysis may seem like another way for outsiders to judge whether Native people's claim to their homelands are legitimate, or not.

In practice, however, you don't need to date human bones to date human remains - you can date artifacts, charcoal, soil, or shell from in or around a burial to get an accurate date range on an interment. DNA analysis is nice to have, from my Euro-American/scientific point of view, but I think it's more important to empower tribes to make that decision for themselves. We as archaeologists need to earn the trust of tribal communities that is required for them to feel comfortable with DNA analysis and also give them the opportunity to 'own' the data, rather than having it exclusively in the hands of outsiders.

Just add a couple things to what was already said, many first Nations or native American histories refer to extensive intermarriage and travel between distant communities with the assumed and result that genetic ancestry does not necessarily correlate to linguistic ancestry. In the Pacific Northwest there are many places where the communities of a town speak a different language than they did 500 years ago, simply because of intermarriage or social connections. Many nations own stories highlight the fact that their first ancestors traveled to distant places and married people from other communities. Even things like cultural shifts evidenced through changing archaeological evidence might not mean displacement of people, simply a displacement of material culture. That is one reason why oral history, material culture, plus DNA evidence and whatever other evidence is available are all important as a whole when trying to get as clear an understanding as possible of the past.