Mr Smith was replaced as shadow Northern Ireland secretary after calling for a second European Union referendum.
He said he had been sacked for his views on the “damage” Brexit would do to the UK’s economy and the Good Friday Agreement.
In an apparent message to Mr Corbyn, he added: “Those views are shared by Labour members and supporters and I will continue to speak up for them, and in the interest of our country.”
Mr Smith has been replaced by Rochdale MP Tony Lloyd.
The Labour leader said Mr Lloyd “is a highly experienced former Government minister who is committed to ensuring that peace in Northern Ireland is maintained and helping to steer the devolution deal back on track”.
Mr Smith who challenged Mr Corbyn for the party leadership in 2016, said he would continue to “speak up” for what he called his “long-held views” on Brexit.
He had written an article in the Guardian urging his party to reopen the question of whether Brexit was the right thing for Britain.
“Given that it is increasingly obvious that the promises which the Brexiters made to the voters, especially but not only their pledge of an additional £350m a week for the NHS, are never going to be honoured, we have the right to ask if Brexit remains the right choice for the country,” he wrote.
“And to ask, too, that the country has a vote on whether to accept the terms and true costs of that choice once they are clear.”
Mr Corbyn has repeatedly insisted it is not Labour’s policy to offer voters a chance to consider the final Brexit deal at the ballot box.
However, the Labour leader has argued Theresa May would have little choice but to call a general election if she lost the key vote in parliament on the agreement.
Mr Smith said remaining in both the customs union and the single market were the only way of preventing what he warns would be the “hardest ever” border in Ireland.
He said: “The damage a disorderly and ill-thought-out Brexit could do in Ireland is enormous.
“We are often told Brexit threatens to ‘reimpose’ a so-called hard border on the island of Ireland, but that understates the problem because the economic border that a hard Brexit would create on Ireland would be the hardest ever.”