I’m not taking an exam means you are not even attempting to take the exam. Taking an exam is sometimes also called sitting for an exam. I’m not passing an exam means you took the exam, but believe that you will not get enough answers correct in order to pass the exam.

Taking an exam means participating / doing an exam, without knowing the outcome yet. Passing an exam means that your result meets the standards, that it was good enough, that your grade is good enough. Not passing means, it wasn't good enough and you have to do it again (if there is this opportinity)