It depends upon the program. Some require that you complete it. Others will not. Furthermore, if you wish to determine your eligibility for loans, then you must also complete the FAFSA. Some states have early (March/April) FAFSA deadlines; however, even if the date has passed, then you can still submit the FAFSA. Early deadlines tend to be for individuals interested in grants (which you are ineligible for as a grad student) and other undergrad scholarships, so nothing really to be concerned about for grad school.
I believe that I did not complete the FAFSA until June for the current academic year, and at other times prior to that, I did not complete it until August, right before the semester started, so I suspect that you are fine.