Is there a closed-form expression for $\prod_{n=1}^{\infty}(1-\frac{x}{n^3})$?

As already noticed in comments, the expression can be obtained from the infinite products for $\Gamma$ (either Euler's one, or Weierstrass's one): $$\Gamma(1+z)=\prod_{n=1}^\infty\frac{(1+1/n)^z}{1+z/n}=e^{-\gamma z}\prod_{n=1}^\infty\frac{e^{z/n}}{1+z/n},$$ and the "algebraic" $1-x/n^3=(1-x^{1/3}/n)(1-e^{2\pi i/3}x^{1/3}/n)(1-e^{-2\pi i/3}x^{1/3}/n)$, giving $$\prod_{n=1}^\infty\left(1-\frac{x}{n^3}\right)=\Big(\Gamma(1-x^{1/3})\Gamma(1-e^{2\pi i/3}x^{1/3})\Gamma(1-e^{-2\pi i/3}x^{1/3})\Big)^{-1}.$$ This easily applies to more general "rational infinite products", as outlined here.