Let $X$ be a complex projective variety, and let $L$ be a line bundle on $X$ of order $r$, that is $L^{\otimes r}\simeq \mathcal{O}_X$.
Question: Is it true that as a principal $\mathbb{C}^*$-bundle, $L$ has reduction of its structure group to $\mu_r:=\{\xi^r=1\}\subset\mathbb{C}^*$? In other words, can $L$ be viewed as a $\mu_r$-bundle?
I think one direction is OK: if $L$ is a $\mu_r$-bundle, then its transition functions $g_{ij}$ take values in $\mu_r$, and hence the $r$-tensor power will have transition function $g_{ij}^r=1$.
But I'm having difficulty seeing the other direction; namely if $L^r=\mathcal{O}_X$ then it is a $\mu_r$-bundle. Any help would be appreciated.
It depends on which topology you use.
Line bundles on $X$ are classified by $H^1(X,\mathbb{G}_m)$, so line bundles with the property $\mathcal{L}^{\otimes r}\simeq \mathcal{O}_X$ are classified by $$ \mathrm{Ker}(H^1(X,\mathbb{G}_m)\xrightarrow{r}H^1(X,\mathbb{G}_m)). $$ On the other hand, $\mu_r$-torsors on $X$ are classified by $H^1(X,\mu_r)$, and a line bundle $\mathcal{L}$ arises from a $\mu_r$-torsor if and only if $[\mathcal{L}]\in H^1(X,\mathbb{G}_m)$ is in the image of $$ H^1(X,\mu_r)\to H^1(X,\mathbb{G}_m). $$ Therefore your question is equivalent to whether the sequence $$ H^1(X,\mu_r)\to H^1(X,\mathbb{G}_m)\xrightarrow{r}H^1(X,\mathbb{G}_m) $$ is exact.
In the analytic or étale topology, the answer is YES. It follows from the exact sequence of (analytic or étale) sheaves called the Kummer sequence: $$ 1\to \mu_r\to \mathbb{G}_m\xrightarrow{r} \mathbb{G}_m\to 1\quad\text{(exact)}. $$ However, in the Zariski topology the answer becomes NO; the exactness on the right of the Kummer sequence is no longer true. The sheaf $\mu_r$ is (non-canonically) isomorphic to the constant sheaf $\mathbb{Z}/r\mathbb{Z}$, so it is flasque in the Zariski topology and hence $H^1(X,\mu_r)=0$. Therefore non-trivial line bundles can never be reduced to a $\mu_r$-torsor.