The codegree Turán density $π_{\text{co}}(F)$ of a $k$-uniform hypergraph (or $k$-graph) $F$ is the infimum over all $d$ such that a copy of $F$ is contained in any sufficiently large $n$-vertex $k$-graph $G$ with the property that any $(k-1)$-subset of $V(G)$ is contained in at least $dn$ edges. The problem of determining $π_{\text{co}}(F)$ for a $k$-graph $F$ is in general very difficult when $k \geq 3$, and there were previously very few nontrivial examples of $k$-graphs $F$ for which $π_{\text{co}}(F)$ was known when $k \geq 4$. In this paper, we prove that $C_\ell^{(k)-}$, the $k$-uniform tight cycle of length $\ell$ minus an edge, has vanishing codegree Turán density if and only if $\ell \equiv 0, \pm 1 \pmod{k}$ when $\ell \geq k + 2$. This generalises a result of Piga, Sales and Schülke, who proved that $π_\text{co}(C_\ell^{(3)-}) = 0$ when $\ell \geq 5$. The method used to prove that $π_\text{co}(C_\ell^{(k)-}) = 0$ when $\ell \equiv \pm 1 \pmod{k}$ and $\ell \geq 2k - 1$ in fact gives a rather larger class of $k$-graphs with vanishing codegree Turán density. We also answer a question of Piga and Schülke by proving that another family of $k$-graphs, studied by them, has vanishing codegree Turán density.