There are a couple of instances of dramatic irony in Act III of Romeo and Juliet, like in scene ii when Nurse comes in carrying the requested cords then begins wringing her hands and lamenting death and Romeo in one breath thus leading Juliet to think that it is Romeo who is dead. But the most significant instance of dramatic irony occurs in scene i when Romeo attempts, for the love he bears for Tybalt, to break up the fight between Mercutio and Tybalt.
Before discussing this further, let's make sure to define dramatic irony and distinguish it from situational irony, which also occurs in Act III. Dramatic irony is one of the three forms of literary irony: verbal irony, situational irony and dramatic irony. Irony, in general, is when what is said or what occurs or what is known is in some way different from and in opposition to what is expected.
For example, if someone steps on your mislaid glasses, it is not expected that you would say "Thank you for your help on that"; saying this would be verbal irony: not expected and the opposite of what is meant. Another example is a fireman who is a pyromaniac: this situational irony is not expected and painfully oppositional to what is needed. A last example is a character who opens a door not knowing that his friends have balanced a bucket of poster paint on it; the audience/reader knows, but the other character does not know of this dramatic irony, which is when the audience/reader knows something that some character(s) does not know.
In Act III, scene i, we know that Romeo and Juliet are married (obviously Romeo knows) but no one else does. Romeo comes on the scene of the antagonism between Mercutio and Tybalt, who is really only looking for Romeo and wishing for a duel with him for his offence of intruding on the Capulet party. Romeo refuses to take up a quarrel with Tybalt and tries his best to quell the mounting, then exploding argument between Mercutio and Tyblat. The reason Romeo gives is that he loves Tybalt better than Tybalt can possibly know.
I do protest, I never injured thee,
But love thee better than thou canst devise,
Till thou shalt know the reason of my love:
And so, good Capulet,--which name I tender
As dearly as my own,--be satisfied.
This scenario--Romeo and the audience knowing what others, especially Tybalt, do not know--is dramatic irony. It differs from situational irony because it involves select foreknowledge and relates to the kind of dramatic effect the scene in the play/novel has upon the audience/reader.
Situational irony can also have a dramatic effect but of a different kind. To distinguish dramatic from situational, an example of situational irony in Act III, scene i is the situation between Mercutio, Benvolio and Tybalt. Benvolio tries to get Mercutio to vacate the streets to avoid a pending sword fight with the Capulets. Mercutio accuses Benvolio of being quick to pick a fight while saying he himself is not like that, thus need not flee. Tybalt enters looking for Romeo. Mercutio immediately tries to incite a fight. The dramatic effect of this situational irony is that we dread what will happen in and as a result of the whole situation while in dramatic irony we have foreknowledge and dominantly dread what will happen to the principal character(s).