Twitter said it was limiting the article's dissemination due to questions about "the origins of the materials" included in the article.
Republicans were outraged by what they called partisan censorship.
Trump, who trails Biden in polls 20 days before the presidential election, blasted the two social media giants.
"So terrible that Facebook and Twitter took down the story of 'Smoking Gun' emails related to Sleepy Joe Biden and his son, Hunter, in the @NYPost," Trump posted on Twitter.
"It is only the beginning for them. There is nothing worse than a corrupt politician."
So terrible that Facebook and Twitter took down the story of “Smoking Gun” emails related to Sleepy Joe Biden and his son, Hunter, in the @NYPost. It is only the beginning for them. There is nothing worse than a corrupt politician. REPEAL SECTION 230!!! https://t.co/g1RJFpIVUZ
The New York Post said the computer had been left by Hunter Biden at a Delaware computer repair shop in April 2019.
The unidentified shop owner told the newspaper that after the computer seemed to have been forgotten, he copied the hard drive and gave the machine to federal authorities.
The shop owner passed the hard drive copy to Rudy Giuliani, President Donald Trump's personal lawyer, who provided it to the newspaper.
While the Biden campaign did not deny the existence of the computer or validity of the emails on it, Giuliani has a record of dispersing disinformation about both Bidens and Ukraine.
In September the US Treasury said one "source" Giuliani met with several times, Ukrainian politician Andrii Derkach, "has been an active Russian agent for over a decade."
The Post criticised Facebook and Twitter for helping Biden's election campaign, saying that no one has disputed the story's veracity.
"Facebook and Twitter are not media platforms. They're propaganda machines," it wrote in an editorial.
Republican Senator Josh Hawley, in a letter to Facebook chief executive Mark Zuckerberg, said the "seemingly selective" blocking "suggests partiality on the part of Facebook."
The Post report, he said, is "clearly relevant to the public interest" and reveals "potentially unethical activity by a candidate for president."
"Twitter's censorship of this story is quite hypocritical, given its willingness to allow users to share less-well-sourced reporting critical of other candidates," Republican Senator Ted Cruz said in a letter to Twitter CEO Jack Dorsey.
The story revived criticisms from the past two years that Joe Biden, when he was in charge of the Obama administration's Ukraine policy, took actions to help his son and the Ukrainian energy company whose board Hunter Biden sat on, Burisma.
Biden has repeatedly rejected such allegations. "I have never spoken to my son about his overseas business dealings," he said flatly in September 2019.
The story focused on one email from April 2015, in which a Burisma board advisor named Vadym Pozharskyi thanks Hunter for inviting him to a Washington meeting with his father.
But there was no indication when the meeting was scheduled or whether it ever happened.
"We have reviewed Joe Biden's official schedules from the time and no meeting, as alleged by the New York Post, ever took place," the Biden campaign said.
Trump's campaign, battling to overcome a steep polling deficit to Biden ahead of the election, quickly issued a campaign statement saying the emails were proof Biden "lied to the American people" about his son's business dealings.