Overtwisted disks lift to overtwisted disks, so if $(M^3, \eta)$ is overtwisted then so is any cover.
The reverse is not true. There are tight contact structures with finite covers that are overtwisted. If the universal cover is tight, the contact structure is called universally tight. There may be simpler references, but for examples of universally tight and virtually overtwisted contact structures, see Ko Honda, On the Classification of Tight Contact Structures II. J. Differential Geometry Vol 55, Number 1 (2000), 83-143.