At the request of the question-asker I am converting my comments to an answer.

In general, for any semisimple Lie algebra $\mathfrak{g}$, is $\mu$ is a (dominant, integral) weight of $\mathfrak{g}$ and $\nu$ is another weight, then the $\nu$ weight space in the irreducible representation $V^{\mu}$ is nonzero if and only if $\mu-\nu$ is a nonnegative integral combination of simple roots.

This means $V^{\mu}$ has a nonzero $0$-weight space if and only if $\mu$ is a nonnegative integral combination of simple roots. Since $\mu$ is dominant (and the cone spanned by the simple roots is contained in the cone spanned by the fundamental weights), the "nonnegative" part is automatic. So what matters is whether $\mu$ belongs to the $\mathbb{Z}$-span of the simple roots, or in other words, the root lattice.

So to check for a particular $\mu$ if it satisfies this condition, you need to ask if it is an integral combination of the simple roots $\alpha_i$, which means you need to understand how the $\alpha_i$ are expressed in the fundamental weights $\pi_j$.

In Type A at least, these coefficients are the columns of the Cartan matrix. For instance, in $\mathfrak{sl}_3$ we have $\alpha_1=2\pi_1-\pi_2$ and $\alpha_2=-\pi_1+2\pi_2$. In other types the same should be more-or-less true, but there might be an issue with duality, i.e., roots vs. co-roots. Since Type A is simply laced, no issue like that arises.

All of this should be in any standard source on the representation theory of simple Lie algebras, e.g., Bourbaki or Humphreys.