Of course, abx is completely correct in saying that the truth of the Hodge conjecture is not a birational invariant.  That said, something slightly weaker is true:  if $X$ and $Y$ are $K$-equivalent, then the Hodge conjecture is true for $X$ if and only if it is true for $Y$.  

Here we say two smooth projective varieties $X, Y$ are $K$-equivalent if there exists a third smooth projective variety $Z$ and birational morphisms $f: Z\to X, g: Z\to Y$, such that $f^*\omega_X\simeq g^*\omega_Y$.  For example, birational Calabi-Yau varieties satisfy this property. The theory of motivic integration then implies that $[X]=[Y]$ in the Grothendieck group of varieties, $K_0(\text{Var})$.  

But now, [this paper of Donu Arapura and Su-Jeong Kang][1] shows that the truth of the Hodge conjecture for $X$ depends only on its class in $K_0(\text{Var})$.  

So the bottom line is:  no, the Hodge conjecture is not a birational invariant in general.  But it is for Calabi-Yau varieties, and it is a "$K$-equivalence invariant."


  [1]: https://doi.org/10.1307/mmj/1163789917