Simple answer: nope. I know this because I've made open source software and I don't have a degree.
The degree isn't required because the knowledge of programming required to make software doesn't change between proprietary and open source software. You still need to able to write good working code, and that doesn't require a degree - you can teach yourself.
Of course, a degree is always beneficial because it'll teach you things you didn't know and often will give you an overview of the business side of things too, but it's by no means necessary.