Irony is used through the speech of the characters and symbols. Dramatic irony is when the reader knows something the character does not. In this case, the reader knows how silly the rivalry is. Symbols can be ironic. For example, when the women visit the ruins in Rome, they represent their relationship but also how each feels about the other. Another example is that both women, though rivals, have much in common.
A few years later, and not many months apart, both ladies lost their husbands. There was an appropriate exchange of wreaths and condolences, and a brief renewal of intimacy in the half shadow of their mourning...
They could have been more than a polite comfort to each other. It is ironic because they would be better off as friends than rivals since they have so much in common. This is ironic because it is obvious to everyone else but them.