The Duke says:
"When remedies are past, the griefs are ended
By seeing the worst, which late on hopes depended."
In this instance the Duke is advising Desdemona's father, Brabantio, on the occasion of his accusation against Othello that he had abducted Desdemona and had used witchcraft or potions to subdue her. Brabantio had expressed his disgust at the notion that his daughter could possibly have willingly gone with someone, he assumed, she feared.
Othello and Desdemona were called to give testimony and both assured their audience that their elopement was a mutual decision, borne out of their love for each other. Desdemona told her father that she owes Othello her allegiance, just as her mother had shown allegiance to him (Brabantio).
Brabantio begrudgingly gives his daughter to Othello saying:
"I here do give thee that with all my heart
Which, but thou hast already, with all my heart
I would keep from thee."
He expresses his disappointment and the fact that he feels betrayed by telling Desdemona:
"For your sake, jewel,
I am glad at soul I have no other child:
For thy escape would teach me tyranny,"
The Duke sees that Brabantio is utterly distraught and tries to advise him. He tells him that when all remedies have been used, our grief should be forgotten. Even if we are faced with the worst and most unpleasant outcomes, where our hopes and expectations have been dashed, we need to move on.
There is no purpose in crying over past grievances for this would only lead to greater pain and trouble:
"To mourn a mischief that is past and gone
Is the next way to draw new mischief on."
Brabantio, however, is not satisfied and insinuates that it is easy for the Duke to give such advice, for he will not have to suffer both "the sentence" and "the sorrow" (of his daughter's treason). He later informs Othello that if she could deceive her father, she might just as easily deceive him:
"Look to her, Moor, if thou hast eyes to see:
She has deceived her father, and may thee."
He vindictively plants a pernicious seed in Othello's mind.