My PhD is in Music Composition, but it was a heavily theory-focused program. I also have many theorist colleagues.
Your question is interesting, and difficult to answer in total detail without writing a book, so I won't try to be exhaustive. Let me first say that the understanding of "Music Theory" is most definitely not complete, and that there absolutely is continuing research.
A lot of people use the phrase "Music Theory" as synonymous with "Theory of Common-Practice Tonal music of the 17th–19th Centuries in Europe," but it the actual field is vastly larger than that. There are so many different genres and types of music throughout all human eras and virtually everywhere on the globe. Even a study of just the music happening throughout the world right now on September 10th would be a massive study requiring a huge number of sub-disciplines of theory, composition, musicology, history, ethnomusicology etc. Historical research involving fragments of music notation from Ancient Greece and from Assyria and Babylon is ongoing and complex. Academic study of the musical structures and cultural impacts of pop, rock, hip-hop and other recorded forms of music is a fairly new and exciting field (for example, although my primary compositional world is centered on various kinds of "atonal" chamber music, my dissertation was a close analytical reading of several albums by a band called Nine Inch Nails). Ethnomusicological studies of Brazilian music and Hindustani music are being taught at my school, and new insights are being made constantly.
But even if we restrict ourselves to notated music more-or-less within the lineage of European "classical" music, theory is far from complete. Composers at the turn of the 20th-century in Europe and America began exploring new harmonic and melodic pitch combinations that are almost totally different than those favored during the common-practice era. They weren't just throwing random notes together, however (until later composers like John Cage occasionally experimented with that too!), they were developing new ideas about how to construct music with its own internal logic. For many composers of a theoretical bent, such as Arnold Schoenberg, this lead to definitions of entirely new theoretical systems. They were primarily interested in composing, but this was simultaneously "research" in new music-theoretical areas. For example, a new theoretical system called "set theory" developed as a way to, among other things, classify and catalog every possible combination of pitches in the 12-TET system. And although that set-theoretical cataloging process is complete, research into the effects of different set combinations, and the musical impact of different set choices, and the ways that one set can be transformed into another set (a sub-branch called Transformation theory) are very much open areas. In fact, I'm almost certain that research along these lines and others is a fundamentally open field that will never be truly exhausted.
But even if we restrict ourselves to only common-practice music of the 18th-century and Europe, there is plenty of ongoing research both historical and theoretical. I think it's fair to say that the underlying basics of common-practice music theory are essentially set and complete by now. The genome has been sequenced, if you will. But research into the specifics of how a particular composer realized those theoretical defaults in particular pieces is essentially inexhaustible. Just because the majority of, say, Beethoven's music has been analyzed, doesn't mean that people aren't finding new ways of looking at his musical choices or previously unexplored measures every year.
I'll try a mathematical analogy. In a sense, the field of polynomial algebra is complete once we've expanded the concept of "number" to include the complex numbers. We can say precisely how many solutions any given polynomial equation will have and confidently state that all of those solutions exist within the complex numbers. However, actually finding the solutions to, for example, a 12th-order polynomial is an entirely different matter. And even if we discover a full-proof, finite method for solving all polynomials of every possible degree, that still doesn't mean anything about whether further research in numbers beyond the complex numbers such as Quaternions and Octonions is worthwhile. And even if it did, that still wouldn't mean there wasn't more research to be done in calculus, or field theory, or statistics, or any other field of mathematics.
Research in a field as vast as Music Theory will never be complete, because we can always discover or create new vistas to explore.