I've interestingly been asked this a number of times in my academic consulting business.
You NEED it for loans, but since you aren't getting federal grant money the timing isn't as critical because loans are not based on need. The exception is if your med school uses fafsa for determining (typically need based) school scholarships they offer and have a deadline. Your med school would likely have let you know about such already or have info in their matriculation packets.
You fill it out after completing your taxes or potentially based on parent's taxes if they are claiming you and you didn't work. If something changed in your income you can note that but for loans it doesn't really matter. If you need to amend your taxes not a big deal, you can also amend fafsa, but it is easier to complete it when taxes are completely done, which is April 15th unless you get a tax extension. If you owe on this year's taxes not a big deal, as you file and set up a payment plan.
You don't have to fill it out right away but should have it done in April or May at the latest for loans, just to avoid any last minute problems.
Some challenges can make you ineligible for loans: certain criminal records, legal citizen status issues, tax fraud, Medicare/Medicaid fraud, etc but then you'd likely already know about that and end up not in med school to begin with.