Firstly, the irony lies in the fact that Lord and Lady Capulet both believe that their daughter has died whilst we, the audience, know that she is only in a state of unconsciousness - a sleep so deep that she seems dead. She had imbibed a potion given to her by Friar Laurence as part of their plan for her to be with Romeo, who was to fetch her later from the family burial-vault after she had been interred.
Furthermore, Lord Capulet's assertion that death has now become his son-in-law adds to the irony, for the reason expressed above.
Further irony lies in the fact that Lord Capulet, addressing Paris says:
"... the day before thy wedding day
Hath Death lain with thy wife."
The audience knows that Romeo and Juliet had already been married in Friar Laurence's cell and Juliet could therefore not be Paris' wife. Nor could she become his, since she is already betrothed.
More irony lies in the fact that Lord and Lady Capulet's grief at this particular moment is not warranted, but they will experience even greater (and truer) grief at the end when they discover the true tragedy of their daughter's demise: that she had killed herself because of Romeo's death, but more, that she was lead to these actions because of the strife between the houses Montague and Capulet, something for which they were responsible. In this sense then, they were indirectly responsible for her death - in that lies the greatest irony.