The most appropriate way to answer this question would be to take a particular staging of the play and consider how that production's staging affects the plot. However, it is possible to consider some moments in the play in which staging might affect what we understand as the plot simply from reading the play. The moments that occur to me most readily are the two moments in which Don John tries to convince Claudio that Hero is unfaithful.
The first is in Act II, scene i. There is a masked dance underway, and different couples are conversing. Much about this scene must be decided by the company producing the play: Are men and women wearing masks or only the men? What sort of dance are they doing? How is the dance performed; are all dancers onstage together? It is just this sort of scene in a Shakespeare play that reminds the reader that the play can't really be understood completely unless it is seen live, as it was intended. In this scene, Hero and Don Pedro are dancing together, as Don Pedro has promised Claudio that he will disguise himself as Claudio to win Hero's hand in marriage.
As the dancers dance themselves off-stage, the non-dancers -- Claudio, and Don John and one of his henchmen -- are left onstage. We do not know, simply from reading, how Don Pedro and Hero have acted together and if Hero is flirting with a man she believes is Claudio or not, but Don John convinces Claudio that Don Pedro has wooed Hero for himself. The plot is concerned here because it could be staged either way. Hero could be presented as somewhat of a flirty girl who is enamoured of Don Pedro or not. And this would affect the plot because it would either give Claudio cause to be jealous or not. The question, for the time being, is cleared up when, later in the scene, Don Pedro presents Hero to Claudio as his fiancee. Don Pedro has indeed wooed in Claudio's name as promised.
The other scene that concerns Hero's fidelity is the scene written by Shakespeare to happen offstage, the wooing of Margaret by Borachio in the name of Hero. It would come at the end of Act III, scene ii. This scene is sometimes presented in dumb-show even though Shakespeare did not include it in his play. Showing the audience what it is that Claudio sees can definitely change the plot for the audience since they can either agree or disagree, based on what they see, about whether Claudio is justified in accusing Hero of infidelity or not.
Both of these moments in which Claudio is led to believe that Hero is not faithful to him could alter the perception of the plot of the play based upon how they are staged.
The essay in the link I've provided below on appearance versus reality in the play gives some more suggestions as to moments where staging affects plot.