Do mutiny and mutation have a common root?
While they have separate roots, it's worth noting a couple of points.
First, among the obsolete senses of "mutation" noted by the OED is "revolution" or "civil insurrection". According to the OED, this sense of "mutation" may be at least partly a borrowing from French "meutation", "meute", which comes from Latin "movere":
In sense ‘revolution’ (see sense 1b) perhaps partly after Middle French meutacion riot, revolt (early 15th cent.) < meute uprising
The similarity of movo, movere and muto, mutare is noted by the Dictionnaire Étymologique de la Langue Latine (A. Ernout & A. Meillet, 1932), but they ultimately dismiss any etymological link:
L'idée de changement est inséparable de celle de mouvement et les sujets parlants ont souvent associé muto à moveo; de là des emplois comme ceux qu'on rencontre dans Plaute, Am. 274 ... et dans les langues romanes; cf v. ital. mutare "voyager", fr. remuer, etc. à côté de muer "changer [de peau]" etc.); ... Ces emplois et ce sens ont donné lieu à l'étymologie mouitare > mutare "mouvoir fréquemment, déplacer", puis "changer". Mais, d'une part, le fréquentatif de mouere est motare et, d'autre part, le sens premier de mutare est bien "changer", comme le prouvent le dérivé mutuus et les composés commutare, permutare; et la forme commoetacula enseigne que l'u de mutare est issu d'un ancien oi.
In other words:
The idea of change is inseparable from that of movement, and speakers have often associated muto with moveo; whence such uses as we meet in Plautus, Am. 274 ... and in Romance languages; cf. Old Italian mutare "travel", French remuer "move", etc. alongside muer "change [one's skin]" etc. ... These uses and this meaning have given rise to the etymology mouitare > mutare "move frequently, move" then "change". But first of all, the frequentative of mouere is motare, and secondly, the original sense of mutare is indeed "change", as proved by the derivative mutuus and the compounds commutare, permutare; and we learn from the form commoetacula that the u of mutare derives from an archaic oi.
The Online Etymology Dictionary traces both words back to different Proto-Indo-European roots. The word "mutiny" goes back to a root "meue" meaning "to push away." The word "mutate" goes back to a root "mei" meaning "to change, move, or go." The two meanings are related in that both involve motion or change, and there is similarity in pronunciation. It is plausible, therefore, to speculate that ultimately the two words are both derived from a common root. But that speculation has no historical evidence. A common root, if there is one, precedes the development of Proto-Indo-European. For at least the last 4000 years, the words have had different histories.