Nothing. He’s not swearing under oath or testifying under penalty of perjury. You can lie all day like a hound dog. Unless your under oath, there’s not much anyone can do.
There is one thing that the press can do if they have the guts. Every time Carney tells a fib, they should burst into laughter and mock him. He lies only work because few, if any, challenge him at the moment he utters the lie.