For a complex algebraic variety X embedded inside a smooth variety Y, the local cohomology sheaves of X in Y carry additional structure of a (mixed) Hodge module. In the hypersurface and the local complete intersection (lci) case, this has been widely leveraged to prove various results about higher Du Bois and higher rational singularities, among other things. We investigate these local cohomology sheaves when X is a toric variety (which is typically non-lci) and prove various results about them. A few applications include showing that the local cohomological dimension of a toric variety is NOT a combinatorial invariant, and some new results about the singular cohomology of toric varieties. This is based on joint work with Hyunsuk Kim.