Do you have to? No.
You could do this under the radar, especially if the amounts concerned were low.
IMO, your biggest concern is personal liability is if someone gets hurt in the gym. Having a formal corporation would shield you personally from any lawsuits. It also means you need to file paperwork, pay taxes and so on, which tends to make a little venture like this become more trouble than its worth.
On top of that, from what I've heard via friends, running a gym is a slim profit margin business in the first place. I wouldn't try to build up a big referrals business. This sounds more like a "beer money" business. Set up the gym that YOU want, and charge people $100/year to use it. Cash only, skip the referrals thing. If someone refers 10 people, then at the 10th person offer them a "lifetime" membership for free. I'd probably also have them sign some sort of liability waiver and post signs "work out at your own risk". That won't prevent you from geting sued, but it will hopefully limit your losses to just the legal fees.