Weinstein's lawyers blasted the judge's comments as "prejudicial and inflammatory", and raised questions about his impartiality. Burke has not ruled on the request.
Burke noted that he has repeatedly warned Weinstein to not use his phone in court, but his exasperated warning on Tuesday - in which he asked Weinstein "Is this really the way that you want to end up in jail for the rest of your life, by texting and violating a court order?" - prompted Weinstein's attorneys to seek his recusal, for the scolding and for a number of other rulings adverse to the defense.
Judge Burke has not yet responded. Weinstein's lawyers said the film producer, who has been seen clutching an iPhone, was using his cellphone before court was in session. "Faced with extreme and unfairly prejudicial negative publicity both pre-trial and now during jury selection, this Court has refused the defendant's requests for additional necessary procedural safeguards".
The 30 potential jurors who were not dismissed Wednesday are expected to return January 16 for the next phase of jury selection.
Los Angeles prosecutors charged Weinstein on Monday with raping one woman and sexually assaulting another in 2013.
But Judge James Burke denied the request, saying there was too much uncertainty over whether Allred would take the stand to prevent her from observing testimony.
Weinstein, 67, has pleaded not guilty to charges of assaulting two women in NY, and faces life in prison if convicted on the most serious charge, predatory sexual assault. If convicted, he could receive a life sentence.
The former studio boss behind such Oscar-winning movies such as Pulp Fiction and Shakespeare in Love has said any sexual activity was consensual.
Los Angeles district attorney Jackie Lacey said the timing of the charges was unrelated to the NY trial. Weinstein has maintained his innocence and has pleaded not guilty to his charges. He has not entered a plea in that case, which will be tried later.
Burke also praised the media and suggested they haven't appeared biased against Weinstein. It is, however, not confirmed if Theron, Hayek and Perez will be called in for the final hearing from a list of 90 potential witnesses.
About 120 prospective jurors are being summoned to court each day.
Harvey Weinstein, followed by his attorney Donna Rotunno (right) today.
"Ms. Allred should not be allowed to sit in court, watch that testimony and share that testimony with her clients", argued one of Weinstein's lawyers, Damon Cheronis.
On Tuesday morning, Judge Burke called out Weinstein for texting on his cell phone while he was seated in the gallery and waiting to take his place at the defense table.