Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Turán number of $\mathcal{F}$, denoted by $ex_r(n,\mathcal{F})$, is the maximum number of hyperedges in an $\mathcal{F}$-free $n$-vertex $r$-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický, Liu and Palmer [Electron. J. Combin. 20 (2013)] determined the Turán number $ex(n,F)$ of a star forest $F$ for sufficiently large $n$. Recently, Khormali and Palmer [European. J. Combin. 102 (2022) 103506] generalized the above result to three different well-studied hypergraph settings, but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general hypergraph star forests.