The exchange between Fauci and Paul was reported in "‘If anybody is lying here, senator, it is you,’ Fauci tells Sen. Paul in heated exchange at Senate hearing" from CNBC.
The formal referral was reported in the news story "Rand Paul sends official criminal referral on Anthony Fauci to DOJ" from The Hill
An October 2021 Washington Post editorial "What the fight between Anthony Fauci and Rand Paul is really about" discussed the accusations from both men but did not mention any further investigation by the DoJ.
The October 2021 Rolling Stone story ["‘Molecularly Impossible’: Fauci Blasts Rand Paul for Covid Lab Theory"], also discussed the accsuations by Paul, but made no mention of any DoJ investigation.
The October 25, 2021 story "Fauci disputes NIH-funded research at Wuhan Institute involved gain of function as Rand Paul calls for his firing" by The Insider again mentions the accusations by Paul and the counter-accusations by Fauci, but makes no mention of any DoJ investigation.
In December 2021 The Independant published "Rand Paul says Fauci should be jailed for five years for lying to Congress – something fact checkers insist he did not do". That story quotes Paul as saying:
But he should be prosecuted for lying, but at the very least he should be taken out of his position because I think he cost people's lives through misinformation
and goes on to cite various inaccurate statements by Paul on Covid-related issues. The story does not mention any action taken by the DoJ on the referral, roughly six months after the formal accusation.
In a story from NBC "Fauci says Sen. Paul's attacks 'kindle the crazies' who have threatened his life" dated Jan 11 2022, there is mention of the earlier exchange but still no mention of any investigation being undertaken by the DoJ.
In fact, I can find no mention of the US DoJ ever having opened or conducted any investigation into Fauci lying to Congress. The DoJ is not required to start an investigation for every referral or letter asking it to do so, not to announce its reasons for not starting an investigation. Nor does it publicly announce all investigations that it starts or closes.
The (lack of) evidence available is consistent with the DoJ having simply ignored Paul's letter of referral, but it does not prove that. There could be a non-public investigation in progress.
