In Othello, to what extent do you agree that if Cassio had remained sober, Iago’s plan would not have succeeded?

Iago's plan to have Cassio dismissed would not have worked, yes, but only at that particular point. The sly Iago would definitely have come up with a different approach. He was absolutely intent on having his revenge and nothing would have stopped him. This much is evident from earlier events in the play.

When he and Roderigo, for example, failed to have Othello dismissed by approaching Brabantio and accusing the general of having kidnapped his daughter, Desdemona, Iago quickly came up with an alternative - play on Othello's insecurities. He systematically manipulated the general and led him to believe that his wife was having an affair. The fact that he suggested that her lover was Cassio shows how devious he was. It was a matter of 'killing two birds with one stone' for he despised both men with a passion. 

Iago had no difficulty in winning trust. Practically every character in the play called him trustworthy and followed his advice. He could, therefore, play on their emotions and provide them with guidelines which would neatly fit into his pernicious plan to destroy them. He cleverly used the idea of 'hold your friends close, but your enemies closer' to devastating effect. 

In this context then, the masterful puppetmaster would have been able to steer anyone into doing his bidding, as he did with Emilia who actually stole a precious possession, Desdemona's handkerchief, to please him. He did the same with Roderigo by directing the poor lovesick fool into whatever direction he wished him to go with the promise that he would win Desdemona's affection as his reward. 

We furthermore witness Iago's determination when he declares, without even a bit of remorse or irony:

O, sir, content you;
I follow him to serve my turn upon him:

He says this when Roderigo asks him why he continues to be obedient to Othello after the general had refused to appoint him as his lieutenant. He also later states:

In following him, I follow but myself;
Heaven is my judge, not I for love and duty,
But seeming so, for my peculiar end:

He equivocally declares his malevolent intent in this instance. When he decides to punish Cassio for the humiliation that he suffered, since Cassio, an outsider, had been granted the position he so much desired, he is unstoppable. Cassio's dismissal is not enough, he actually wants him killed and uses Roderigo for this purpose. The big prize is, however, the general. He turns him into a blustering, angry and vindictive fool whom he will supposedly help in achieving his revenge.   

Of all the characters whom Iago manipulates, it is ironically only Roderigo who challenges him on realising that the malevolent puppeteer is leading him on. Tragically, Roderigo's enlightenment arrives too late. The scene has already been set for Iago's malevolence to prosper and when he fathoms that Roderigo can jeopardise his master plan, he kills him.

So, therefore, all things considered, Iago's malice and perverse desire for revenge were forces greater than any situation or person to stop him. He would have found some other way in which to carry his malice through to its rancorous conclusion.