I realize that I might be a bit too late the hero, but I'm leaving this comment that got too long for the comment box as an answer.
What seems to have been forgotten by the OP and the answerers is that for one to speak of orthogonality, one should always specify the associated inner product. For the Hermite polynomials $H_n(x)$, the relevant inner product is
$\langle f,g \rangle=\int_{-\infty}^\infty f(x)g(x)\color{red}{\exp(-x^2)}\,\mathrm dx$
Having said this, the general idea in the other two answers is correct: $H_2(x)H_3(x)$ is indeed an odd function, while $\exp(-x^2)$ is even. Their product is odd, and thus $\langle H_2,H_3\rangle$ certainly ought to be zero. As you are now dealing with an improper integral, some more finesse in proving that the integral is sensible is needed; that work is left as an exercise.