The quotients of abelian **surfaces** (over $\mathbb C$) by finite groups are classified by Yoshihara. In particular he determines the possible Kodaira dimensions.
For instance, if the holonomy part of the group ( quotient by its maximal translation subgroup ) has cardinality greater than $24$ then he shows the quotient is rational.

The precise reference is

- Yoshihara, Hisao.
*Quotients of abelian surfaces.* Publ. Res. Inst. Math. Sci. **31** (1995), no. 1, 135--143.

Unfortunately, I am not aware of any electronic version of this paper.

For higher dimensional abelian varieties I am not aware of any work studying the
finite quotients. But for 3-dimensional complex tori there is for instance this paper by Birkenhake, González-Aguilera, and Lange which classifies the possible finite subgroups. The same
authors also have a paper dealing with finite subgroups of the $3$-dimensional abelian varieties (over $\mathbb C$ if I remember correctly).

**EDIT (May 18)**

You may want to take a look at Complex crystallographic groups I and II. The authors study compact quotients $X$ of $\mathbb C^n$ by discrete subgroups $\Gamma \subset \rm{Aff}(\mathbb C^n)$.

In dimension two they obtain classification results for the pairs $(X, \Gamma)$ assuming
$\Gamma$ (more precisely its holonomy part) is generated by reflections (paper I); or $X$ is rational (paper II).