For $ξ\geq 0$ and $d \geq 3$, the higher-dimensional Liouville first passage percolation (LFPP) is a random metric on $ε\mathbb{Z}^d$ obtained by reweighting each vertex by $e^{ξh_ε(x)}$, where $h_ε(x)$ is a continuous mollification of the whole-space log-correlated Gaussian field. This metric generalizes the two-dimensional LFPP, which is related to Liouville quantum gravity. We derive several estimates for the set-to-set distance exponent of this metric, including upper and lower bounds and bounds on its derivative with respect to $ξ$. In the subcritical region for $ξ$, we derive estimates for the fractal dimension and show that it is continuous and strictly increasing with respect to $ξ$. In particular, our result is an important step towards proving a technical assumption made in previous work by the first author and Gwynne. These are also the first bounds on the distance exponent for LFPP in higher dimensions.