As a consequence of the main theorem in
Di Liberti, I, and Loregian F.. "Homotopical algebra is not concrete." Journal of Homotopy and Related Structures (2017): 1-15.
(arXiv free version here), that the homotopy category of "almost all" known model categories is not concrete, I claim that "almost all" model categories produce non-reflective localizations.
Proof. All the examples we give in the paper are concrete categories (as a consequence of Remark 2.2, being concrete is a very weak assumption on a category); if you now assume $Ho(M)$ is a reflective localization of $M$, composing with a faithful functor $M \to Set$, you would get a faithful functor $Ho(M) \to M\to Set$.
This contradicts our Theorem 4.8.
