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Tightness/overtwistedness is pretty hard. One place to start would be the paper Right-veering diffeomorphisms of compact surfaces with boundary I by Honda-Kazez-Matic, which gives a necessary and sufficient condition in terms of "right-veering" mapping classes. However, their result is not algorithmic since it requires checking something on all open books, not just a particular given one. Andy Wand has recently claimed to have an algorithm that detects tightness. I don't believe that there is a publicly available preprint yet, but there is an online talk by Wand and also some nice blog posts by Peter Lambert-Cole here and here.