I did a sibling study on my adult children and found out that my 23 year old son has a different father. We contacted the biological father and he is not willing to do a DNA test. My son birth certificate has my ex-husband name listed as his father. I did live together with my son biological father when my son was two years old in North Carolina, but we never got married. After three years I moved to Texas.
What rights do my adult son have to legally force the issue of establishing paternity, and do he have any inheritance rights, or any legal rights?

You need to contact a family attorney. At this point he may not have any rights.

I did a sibling study on my adult children and found out that my 23 year old son has a different father. We contacted the biological father and he is not willing to do a DNA test. My son birth certificate has my ex-husband name listed as his father. I did live together with my son biological father when my son was two years old in North Carolina, but we never got married. After three years I moved to Texas.
What rights do my adult son have to legally force the issue of establishing paternity, and do he have any inheritance rights, or any legal rights?
If your ex-husband's name is on BS your ex-husband is the legal father and you CANNOT force the biological father to do DNA test.Your son does not have any inheritance rights from his boyfriend.

Normally no rights what so ever. If there is an issue to estate, at probate sometimes they can get a DNA test .