It's not clear what if anything the White House did with that information. A story published by Politico minutes before the Post story says the White House is actively searching for Flynn's replacement.
What the Post article also reveals is that Flynn's deception became a topic of active concern in the final days of the Obama administration. What they knew was that Flynn's public claims about his contacts with the Russian Ambassador were false and suspected that he had misled at least incoming Vice President Pence. Yates, CIA Director John Brennan and DNI James Clapper discussed the matter repeatedly. They recommended briefing the new President so he could decide himself how to deal with the situation. But this: "FBI Director James B. Comey initially opposed notification, citing concerns that it could complicate the agency’s investigation."
When Obama left office on January 20th, Clapper and Brennan resigned their positions, leaving only Yates in place in her caretaker capacity. After Sean Spicer again vouched for Flynn's account on the 23rd, the matter took on renewed urgency and now Comey agreed the White House should be briefed.
It's not clear precisely when Yates and the intelligence official delivered their message to Donald McGahn in the White House. It had to be between January 23rd and 30th, though the Post story suggests it was closer to the latter date.
What is now clear is that the Trump White House has known about Flynn's deception for at least two weeks.
James Comey must have thought he had a real investigation on his hands to want to avoid compromising it, even at the risk of leaving the President in the dark about such a critical matter.
Did Yates message to the White House color the decision to fire her on January 30th?
Why has the White House taken no apparent action for two weeks? As recently as Friday President Trump told reporters he knew nothing about Flynn's deception. As I noted below, the simplest explanation is that Trump knew about the Flynn/Russia back channel all along.