Student authors own the copyright in their dissertations and theses. Unless an advisor materially contributes significant content to a dissertation or thesis, the advisor does not usually own any of the copyright in the work. While advisors are invaluable guides in the process of research and writing, and often contribute significantly to the shape of ideas within a dissertation or thesis, ideas are not protected by copyright; only the form (words, images, music, illustrations, etc.) of the ideas can give rise to copyright ownership.
Degree requirements in a number of schools require students to deposit their dissertation or thesis in the University Digital Conservancy, a center for long-term access and preservation of scholarly materials produced at the University of Minnesota, hosted by the University of Minnesota Libraries. Posting your dissertation or thesis online does not in any way force you to relinquish your copyrights - every work is automatically protected from the very first time it is saved, and just posting something online does not mean it is available for public use.
However, if you are considering posting your dissertation (or are required to submit your thesis) to the Conservancy, there are some serious issues you should consider first, including whether your work contains sensitive data, potentially patentable inventions, is in its current form a candidate for commercial publication, or contains the work of collaborators who may also hold a copyright interest.
You may want to learn more about Fair Use, the copyright doctrine that lets you reproduce limited segments of existing work, without permission or payment, under certain circumstances.
You may also want to explore some common copyright use issues in research.