Jealousy abounds in this play in a variety of characters (and is surprisingly lacking in some characters, such as Don Pedro's lack of jealousy when Beatrice weds Benedick after she has turned down Don Pedro's proposal). Some of the jealousy, however, is hidden under the guise of other emotions.
The jealousy of Claudio is one of the things on which the action of the whole play hinges. When Claudio believes that Hero has been unfaithful to him before their wedding (because of Don John's nefarious plan with Borachio and Margaret) he absolutely explodes with rage
CLAUDIO: Sweet prince, you learn me noble thankfulness.
There, Leonato, take her back again.
Give not this rotten orange to your friend.
She's but the sign and semblance of her honour.(30)
Behold how like a maid she blushes here!
O, what authority and show of truth
Can cunning sin cover itself withal!
Comes not that blood as modest evidence
To witness simple virtue. Would you not swear,(35)
All you that see her, that she were a maid
By these exterior shows? But she is none:
She knows the heat of a luxurious bed;
Her blush is guiltiness, not modesty. (IV.i)
In truth, Claudio's honor has been offended (and he feels he has been tricked and made to look a fool, which is perhaps even worse) but it his jealousy -- that Hero would sleep with another man instead of him before they were married -- is what causes him to react with such violence. He emphasizes that he had treated Hero with the utmost respect.
I know what you would say. If I have known her,
You will say she did embrace me as a husband,
And so extenuate the forehandsin. No, Leonato,
I never tempted her with word too large,
But, as a brother to his sister, showed(50)
Bashful sincerity and comely love.
Claudio is explaining that he does not mean that Hero is impure because he, Claudio has slept with her -- but rather it is the fact that Hero has slept, he thinks, with someone else. He does not say this outright, but he certainly is feeling intense jealousy that another man has had Hero's affections before he has. It causes him to insult his beloved Hero in such a way that it makes her swoon. It is this jealousy and obstinacy which makes the deception of Hero's "death" necessary in the eyes of Beatrice and the Friar.
Don John's jealousy (or perhaps the correct word is envy) is also a major factor in the play. The entire deception which causes the ruin of Hero's character (a woman which Don John has no reason to dislike or want to harm) is contrived by him because he is envious of his brother's legitimacy, high nobility, and power. Don Pedro is respected by everyone because he is the legitimate prince of Aragon, in addition to being a good man and an effective leader, while Don John's life is forever eclipsed by Don Pedro's because of the circumstances of his birth. Don John's innate jealousy makes him want to hurt Don Pedro, and Don Pedro's friends, in any way he can, for any reason. His envy causes him to be rather directionlessly evil, which is why he plans to ruin the lives of two people with whom he has no quarrel whatsoever.
DON JOHN: Come, come, let us thither. This may prove food to
my displeasure. That young start-up hath all the glory of(55)
my overthrow. If I can cross him any way, I bless myself
every way. You are both sure, and will assist me? (I.iii)
Don John doesn't have to have a reason, or even a specific motive to do harm; this is a product of his envy for his brother.