Akshata Murty is reported to have received £11.6m in dividends in the past year from Indian firm Infosys.
But her non-dom status means she is not liable for UK tax on income earned abroad. She would not have paid UK tax, at a rate of 39.35%, on the dividends.
The £30,000 fee is chargeable if a person has lived in the UK for at least seven of the previous nine years.
Under government rules, people can be granted non-dom status - meaning the UK is not considered their permanent home - if they live in the UK but intend to go back to their home country.
Labour leader Sir Keir Starmer said Mr Sunak faced "very serious questions to answer" about his family's finances, adding: "If it now transpires that his wife has used schemes to reduce her tax, while he's been increasing taxes on working people, that's breathtaking hypocrisy."
The Liberal Dems have urged Mr Sunak to ban the partners of ministers from claiming non-dom status, calling it a "loophole".
Ms Murty is an Indian citizen and has retained family ties there, and the BBC understands she has said she would eventually like to return there.
She owns a 0.9% stake in the software firm Infosys - founded by her billionaire father - estimated to be worth more than £500m.
Business Secretary Kwasi Kwarteng defended Mr Sunak, saying it was "completely unfair" to scrutinise the tax affairs of Ms Murty, "who is not a politician" and rejecting opposition claims that she was sheltering from UK taxes.
And 10 Downing Street has dismissed reports that it is behind a series of politically damaging leaks against the chancellor as "baseless" and "categorically untrue."
A non-dom is a UK resident who declares their permanent home, or domicile, outside of the UK.
A domicile is usually the country his or her father considered his permanent home when they were born, or it may be the place overseas where somebody has moved to with no intention of returning.
For proof to the tax authority, non-doms have to provide evidence about their background, lifestyle and future intentions, such as where they own property or intend to be buried.
Those who have the status must still pay UK tax on UK earnings but do not need to pay UK tax on foreign income. They can give up their non-dom status at any time by stating on a tax return that they intend to live in the UK and wish to be considered British for tax purposes.
Ms Murty has chosen to be domiciled in India via her father, the billionaire Narayana Murty, which means she doesn't need to pay taxes in the UK on dividends she receives from her stake in his company.
Ms Murty married Mr Sunak in 2009.
The fee for non-dom status rises to £60,000 a year when a person has been in the UK for at least 12 of the previous 14 years.
And anyone living in the UK for 15 years automatically loses the status.
Ms Murty has declined to say when her non-dom status began.
"India does not allow its citizens to hold the citizenship of another country simultaneously," Ms Murty's spokeswoman said. "So, according to British law, Ms Murty is treated as non-domiciled for UK tax purposes.
"She has always and will continue to pay UK taxes on all her UK income."
But tax experts have questioned Ms Murty's statement, suggesting UK non-dom status is a "choice" and something people can give up.
Asked if Ms Murty paid tax on her Infosys dividend payments in India, her representative said "international tax" was paid on her "international income".
It is not clear, however, whether she pays tax on her dividend income in India or another jurisdiction.
The Indian government says dividend income is taxable at the rate of 20% for non-residents, unless they qualify for any deductions.
But the dividend rate can come down to 10% for people who are eligible to benefit from the UK's tax treaty with India.
In a letter to Mr Sunak, Labour asked if his wife paid all foreign tax in India or in a tax haven such as the Cayman Islands.
The chancellor made the Cabinet Office aware of his wife's tax status as part of his declaration of interests when he first became a government minister in 2018.
In an interview with the Sun newspaper, Mr Sunak said Ms Murty "followed the letter of the law" over taxes.
"To smear my wife to get at me is awful," he added, saying: "And if she was living here and didn't just happen to be married to me, this obviously would not be at all relevant."