The file was uploaded to the official website, but has since been removed.
The Cabinet Office told the BBC: "We apologise to all those affected and are looking into how this happened."
Among the addresses are those of Sir Elton John and former director of public prosecutions Alison Saunders.
Also on the list of 1,097 honours recipients are high-profile names such as cricketer Ben Stokes, former Conservative Party leader Iain Duncan Smith, TV cook Nadiya Hussain, and former Ofcom boss Sharon White.
A government spokesman said: "A version of the New Year Honours 2020 list was published in error which contained recipients' addresses.
"The information was removed as soon as possible.
"We have reported the matter to the ICO [Information Commissioner's Office] and are contacting all those affected directly."
The ICO, which has the power to fine organisations for data breaches, said it will be "making enquiries".
Data rights lawyer Ravi Naik said the government could face legal action from those whose addresses were published, as well as from the ICO.
He also warned that anyone who came across the information should tell the ICO and not pass it onto others - because they themselves might face legal action.
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