BBC reports they were reportedly convicted for promoting corruption, prostitution and propaganda.
The video showed them dancing by Tehran's Azadi (Freedom) Tower.
Authorities are handing heavy sentences to people seen to be involved in protests after the death of a woman who was detained by morality police.
The couple did not link their dance to the ongoing protests in Iran.
A source has confirmed to BBC Monitoring that the couple's arrest came after they posted the video to their Instagram accounts, which have a combined following of nearly two million.
Anti-government protests - labelled "riots" by Iran's regime - swept across the country after Mahsa Amini, 22, died in police custody in September last year. She was arrested in Tehran for allegedly violating the rule requiring women to cover their hair with a hijab, or headscarf.
Astiazh Haqiqi, 21, and her fiance Amir Mohammad Ahmadi, 22, are said to be convicted of "promoting corruption and prostitution, colluding against national security, and propaganda against the establishment".
The family home of Ms Haqiqi, who lists her profession as a fashion designer, was raided before the arrest.
It is unclear how long the sentence is for each of the separate convictions they are facing. They have each been sentenced to a total of 10 and a half years - a combined sentence for the charges.
If their verdicts are upheld, they will have to serve the longest one of those sentencing terms.
According to reports, they were also handed a two-year ban on using social media and leaving the country.
Iran's protest movement that began in September has become one of the most serious challenges to the Iranian regime since it came to power in the 1979 revolution.
To quell the protests, the state has been handing out severe sentences to people involved in the unrest, including executing at least four protesters. earlier this month and in December.
While Mahsa Amini's death was the catalyst for wider unrest in Iran, it has also been driven by long-standing discontent over poverty, unemployment, inequality, injustice and corruption.
Hundreds have been killed and thousands arrested during demonstrations over the circumstances of her death.
Photo BBC screenshot