This is a great question! Remember the definition of dramatic irony: dramatic irony is defined as when someone on the stage and/or the audience knows something that one or several of the other characters do not. One of the essential elements of dramatic irony in this play over all is of course the tragic end, when we know that Juliet is just about to wake up, but Romeo doesn't, and goes on killing himself, just before Juliet revives.
However, when we think about dramatic irony in this famous scene, one place to start is towards the beginning. Remember that Romeo and Juliet have just had rather a sudden and quick meeting before Juliet is dragged away. Therefore Romeo is left wanting to have some confirmation of what he is experiencing and that he is not dreaming.
He receives this confirmation in the form of Juliet appearing on the balcony and addressing "Romeo," ironically not knowing that he is actually there to hear every word. Listen to what she says:
O Romeo, Romeo! Wherefore art thou Romeo?
Deny thy father and refuse thy name;
Or, if thou wilt not, be but sworn my love,
And I'll no longer be a Capulet.
This is enough to give Romeo the confirmation that his feelings are mutual and shared by Juliet, and he ponders in an aside whether he should reveal his presence at this stage or wait to hear more. Therefore this is dramatic irony because we know that Romeo is actually there, hearing every word that Juliet is actually addressing to Romeo, though she doesn't know where he is.