Is the quote “hear me with patience” an example of dramatic irony in Romeo and Juliet?

It is indeed. Dramatic irony can be defined as something the audience knows but that one of the characters in the play doesn't. In this particular scene, Juliet drops a bombshell: she's refusing to go ahead with her wedding to Paris, the man her parents have arranged for her to marry. Understandably, Lord and Lady Capulet are absolutely furious at what they see as their daughter's ingratitude and insubordination. Juliet desperately wants them to try and see it her way, literally begging on her knees for her father to be patient and listen to what she has to say:

Good Father, I beseech you on my knees, Hear me with patience but to speak a word (Act III, Scene v).

But we know, even if Juliet doesn't, that patience is not exactly a characteristic one normally associates with her father. Indeed, it is Capulet's total lack of patience which will lead directly to Juliet's tragic death.