No, your employer is not required to pay you while you take your leave; they only offer unpaid leave. If the employee has incurred PTO and vacation days, they may receive that pay during their time in rehab.
It's best to speak with your employer about the details surrounding FMLA and your unique situation. This is the surest way you will understand exactly what the steps involved are, in partnership with your employer, to approach this stage in your life with diligence and knowledge.