It suffices to prove the statement for the case when $X_{\Sigma}$ is projective, as the general case reduces to this one by taking a suitable refinement of the fan.

The symmetry of the toric local height follows readily from the analogous property for the local height, see Theorem 2.46(1). For the multilinearity, let ${\overline{L}}_{d}^{\prime}$ be a further metrized line bundle. By the moving lemma, there are sections $s_{i}$ of $L_{i}$, $0\leq i\leq d$ meeting properly on $Y$ and $s_{d}^{\prime}$ of $L_{d}^{\prime}$ such that $s_{0},\dots,s_{d-1},s_{d}^{\prime}$ meets properly on $Y$ too. By Theorem 2.46(1),

and a similar formula holds for the canonical metric. By the definition of the toric local height, $\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1},{\overline{L}}_{d}\otimes{\overline{L}}_{d}^{\prime}}(Y)=\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)+\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1},{\overline{L}}_{d}^{\prime}}(Y)$. The inclusion-exclusion formula follows readily from the symmetry and the multilinearity of the local toric height. ∎