Members of the Public and Commercial Services Union will walk out on 19 May.
In a ballot, 96% of the 150 affected members voted for strikes on a 53% turnout. 247 people are employed at the office in St Paul's Place.
BIS said it will provide "comprehensive support" to affected staff.
The government announced plans to shut the office by 2018 in January.
Mark Serwotka, general secretary of the Public and Commercial Services Union, said: "This vote sends the clearest possible message to BIS ministers and senior officials that these plans are wrong and must be reversed.
"The closure of this office is precisely the opposite of what the government claims to want for the North and the wider civil service."
He said the union was also considering a timetable for further walkouts.
Announcing the plans, permanent secretary Martin Donnelly said it was part of a move to create a combined central HQ and policy centre in London and establish six business centres across the UK.
He said the decision was taken in an effort to "modernise the way we work, reduce operating costs and deliver a simpler, smaller department that is more flexible and responsive".
A BIS spokesperson said: "Whenever a proposal such as this is made, people's livelihoods and families are directly affected, and these decisions are never easy. It remains our top priority that all our staff are fully briefed and consulted during the process.
"We take their future and the contribution they have made very seriously, and will provide comprehensive support to all those facing a potential change or loss of job."

Rail electrification is already going ahead in south Wales but no decision has yet been made on the north.
The Wales Office and Welsh government are working with businesses to make the case for the investment needed for the project to get the go-ahead.
But Mr Osborne said a decision would be affected by the HS2 project and a rail hub possibly being created in Crewe.
Electrification means faster and more reliable services which help businesses and their workers who commute.
Welsh Secretary Stephen Crabb also said electrification could be a game-changer and could happen in the longer-term.
Speaking on a visit to Denbighshire, Mr Osborne said: "Lets make the decision first on HS2 because that new rail hub at Crewe will be crucial for north Wales and it's crucial for the investments we're making in the north Wales economy."

Anne sent the eight-line verse to a friend in 1942.
The auction house in the Netherlands had valued the rare, handwritten note at €30,000-€50,000.
Anne is best remembered for her diaries of life as a German-born Jew in occupied Amsterdam in World War Two.
The poem was signed and dated 28 March 1942, which was less than four months before the Frank family moved into a secret apartment to hide from German forces.
Addressed "Dear Cri-Cri", it was given to Christiane van Maarsen, the older sister of one of Anne's best friends, Jacqueline.
The Bubb Kuyper auction house, based in the Dutch city of Haarlem, said Jacqueline van Maarsen was selling the poem because she knew her sister, who died in 2006, did not see it as a prized possession.
In a note published on the auctioneers' website, she wrote, "My sister (nicknamed Cri-Cri) tore this page out of her poezie album and gave it to me around 1970. I know that my sister was not as attached to this verse from Anne to her as I am to the verse Anne addressed to me."
It was common for girls of the era to collate notes from friends in scrapbooks they called poezie albums.
The first four lines of the poem, which encouraged the older sister to work hard at school, are known to have been copied from a magazine.
The second four carried on the same edifying theme, but may be Anne's original work.
The auctioneers noted that Anne switched her style of handwriting midway through writing.
Her diaries have become one the most important documents to emerge from the Holocaust. Her father published them after she died of typhus in the Bergen-Belsen Nazi death camp aged 15.
Civil servants at the Department for Business, Innovation and Skills (BIS) have announced plans to stage a 24-hour strike in protest at plans to shut the department's Sheffield office.
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There is a "really strong case" for the electrification of the north Wales rail lines, Chancellor George Osborne said.
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A poem by Dutch schoolgirl Anne Frank written shortly before her family went into hiding from the Nazis has sold at auction for €140,000 ($148,000; £120,000), well over the estimate.