Mr Khan said he would step down as an MP two weeks ago after he was found guilty of sexually assaulting a 15-year-old boy.
As of yet, the Wakefield MP has not officially resigned from his seat despite being expelled from the Tory party.
Speaking while in Stevenage ahead of the local elections next week, Sir Keir said: “The disgraced Tory MP in Wakefield said he would resign and he still hasn’t actually resigned.
“Wakefield deserves better than that. He should get on with it, so we can have a by-election.”
On April 14, Mr Khan said he would stand down from his west Yorkshire seat following the ruling, but has yet to formally do so.
In a statement, he said: “While legal proceedings are ongoing, I do not believe that it would ordinarily be appropriate to resign.
“However, owing to long delays in the legal process, my constituents have already been without visible parliamentary representation for a year. Even in the best case scenario, anticipated legal proceedings could last many more months.
“I have therefore regrettably come to the conclusion that it is intolerable for constituents to go years without an MP who can amplify their voices in Parliament.
“Representing them has been the honour of my life, and they deserve better than this. Consequently I am resigning as MP for Wakefield and withdrawing from political life.”
To trigger a by-election, Mr Khan must write to the Treasury to confirm his resignation, where he will be placed as a Crown Steward and Bailiff of the three Chiltern Hundreds of Stoke, Desborough and Burnham. The position requires MPs to vacate their seats.
Mr Khan was found guilty of sexual assault at Southwark Crown Court earlier this month.
Now he has apologised to his family and the community for the “humiliation” but he said he plans to clear his name.
His legal team has vowed to appeal against the conviction.