There are many things that characterize a Shakespearean tragedy. However, while we can recognize some of these characteristics in Act 4, Scene 1 of Much Ado About Nothing, because the scene ends in faked death rather than real death with the hopes of a happier ending to come, the scene is more fitting of a comedy rather than a tragedy.
All of Shakespeare's comedies, true to the classic Greek comedies, contain a certain amount of tension and conflict. Also, the lovers in his romantic comedies face conflicts and "must overcome obstacles" in order to be united (Schwartz, "Shakespeare's Plays: Comedy"). True to Shakespeare's comedies, Hero must overcome the obstacle of having her reputation slandered by Claudio and being accused of promiscuousness and unfaithfulness, as we see when he accuses her of knowing "the heat of a luxurious bed" and of being "an approved wanton," meaning "proved whore" (IV.i.38, 42). Therefore, the fact that this scene is full of tension does not alone say that this scene belongs in a tragedy rather than a comedy.
However, there are a few ways in which this scene relates to scenes commonly found in tragedies. Just like the tragic hero, the character Hero undergoes a great deal of suffering in this scene when before she was exceedingly happy, especially happy about her new betrothal to Claudio. We can see Hero's happiness, when we see her whisper in Claudio's ear that "he is in her heart" and again when she gaily refers to Claudio as her "new-trothed lord"(II.i.278-279, III.i.39). Also, like a tragedy, this scene culminates in death, albeit it is the decision to fake her death. Another similarity between a tragedy and this scene is that the tragic hero must have a character flaw. While Hero is completely flawless, Claudio is creating the tragedy in this scene due to is excessive pride and gullibility. Both negative character traits led him to be duped by Don John into misjudging Hero. A third similarity between this scene and a tragedy is that tragedy occurs due to a moral struggle, or a struggle between good and evil. While Hero is wholly good, Claudio is being influenced by the evil, immoral forces we see presented in Don John.