Typical weighted random simplices $Z_μ$, $μ\in(-2,\infty)$, in a Poisson-Delaunay tessellation in $\mathbb{R}^n$ are considered, where the weight is given by the $(μ+1)$st power of the volume. As special cases this includes the typical ($μ=-1$) and the usual volume-weighted ($μ=0$) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of $Z_μ$ satisfies a central limit theorem in high dimensions, that is, as $n\to\infty$. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight $μ=μ(n)$ to depend on the dimension $n$ as well. A number of special cases are discussed separately. For fixed $μ$ also mod-$φ$ convergence and the large deviations behaviour of the logarithmic volume of $Z_μ$ are investigated.