In Much Ado About Nothing, Don John is a villain and a bastard. These archetypes confine him to playing a static role in the comedy. Most bastards are stock characters in literature: they are cultural stereotypes who both fulfill and parody the stereotype.
In Shakespeare's time, a bastard was a dispossessed son: illegitimate, impure, a mongrel, a rogue, an impostor, one who thinks he deserves more than he gets, and jealous of his natural-born brother (Don Pedro).
Benedick knows this:
Even Don John himself admits his own nature:
I cannot hide what I am: I must be sad when I have cause and smile at no man's jests, eat when I have stomach and wait for no man's leisure, sleep when I am drowsy and tend on no man's business, laugh when I am merry and claw no man in his humour.
As much as he tries to be legitimate and social, Don John resents both his status and those who look down on him because of it.
Static characters are usually secondary characters. They are always sort of flat or one dimensional. They do not undergo any real sorts of changes in the course of the work of fiction. Don John really fits with this definition because he is just a bad guy. He does not really do anything in this play other than act like a complete jerk.
Don John does not show any real redeeming qualities in the play. He is nasty and antisocial. He tries hard to ruin his brother's life and he does a terrible thing (for no good reason) when he smear's Hero's name.
Since he spends the whole play being a bad guy, he is static.