Let $\mathfrak{g}$ be a Lie algebra and $P$ its weight lattice. Let $I$ be the set of vertices of the Dynkin diagram of $\mathfrak{g}$. Suppose that $I=\{1,\ldots,n\}$.
Let $\alpha_1, \ldots, \alpha_n$ be the simple roots and $\alpha_1^{\vee}, \ldots, \alpha_n^{\vee}$ be the simple coroots. Let $\omega_1, \ldots, \omega_n$ be the set of fundamental weights. We have $\omega_i(\alpha_j^{\vee}) = \delta_{ij}$.
Let $\lambda \in P$. Are there non-zero weight $\lambda$ such that $\lambda(\alpha_{i}^{\vee})=0$ for all $i \in I$? Thank you very much.
If $\mathfrak{g}$ is abelian, then there are no roots, coroots, or fundamental weights, so only the trivial weight is a linear combination of fundamental weights (Milne 2012, p. 32).
If $\mathfrak{g}$ is semisimple, then every weight that satisfies your condition is trivial (Milne 2012, p. 51).