The 74-year-old and co-star Jim Broadbent filmed scenes at Cartwright Hall in Lister Park this week.
Bradford Council's museums team, who manage the civic art gallery, Tweeted that the building would be closed to the public on Tuesday, Wednesday and Thursday. The upper galleries will also be shut on Friday.
Helen Mirren poses with a hen party and enjoys a curry in BradfordFilm crews and trucks were pictured outside the hall, which was built in the 'Bradford baroque' style in 1904 using a £40,000 bequest from mill baron Samuel Lister. It is home to a large collection of David Hockney's work.
It's the second week Dame Helen and her fellow cast members have been in Bradford for the shoot and filming has already taken place at City Hall.
The Oscar winner is believed to be staying in the historic Great Victoria Hotel, which is known for its 19th-century grandeur.
She posted photos on Instagram with a group of women in 1980s fancy dress partying in the hotel's bar, and also documented her visits to the Alhambra Theatre to watch panto and Indian restaurant Mumtaz, where she enjoyed a curry and posed for selfies with fans.
Dame Mirren also referred to the city as 'magical' in a caption on her social media account.
She is believed to be playing the wife of The Duke's main character Kempton Bunton in the film.
This week, actress and singer Hannah Pitt posted a photo on Instagram of herself in 1960s costume at Cartwright Hall, suggesting she is also part of the cast.
The incredible heist that The Duke is based on
The Duke is set in 1961, when 60-year-old bus driver Kempton Bunton was accused of stealing Francisco Goya's portrait of the Duke of Wellington from the National Gallery in London - the only theft in the institution's history.
He sent ransom notes claiming that he would return the artwork if the government spent more on social care for the elderly and free TV licences.
He claimed he managed to get into the gallery through a toilet window after talking to staff and discovering the security system was deactivated for cleaning during the morning. He later contacted news agency Reuters to request a donation to a charity to fund TV licences for poorer people in return for the painting.
The ransom was declined and the painting was not seen again for four years, until Bunton contacted a newspaper and informed them he would leave the picture in the left luggage office at Birmingham New Street Station.
He was traced and arrested, but at his trial was only convicted of stealing the frame, which was still missing. His barrister successfully argued that he had never intended to keep the painting permanently so it had not been stolen. He was sentenced to three months in prison and the law was subsequently changed as a result of the case.
In 1996 it was first rumoured that another person may have actually committed the theft on Bunton's behalf, but it was not until 2012 that a confidential file was released by the National Archives which stated that in 1969 his son John had confessed to stealing the painting and reiterated that the intention had always been to return it. He said his father and warned him and his brother Kenneth not to come forward. No decision was taken to prosecute John for theft or Kempton for perjury.
Bunton died in 1976.