The only thing your shoes have to do is protect your feet, be comfortable and provide good footing/traction. Beyond that, it's all personal preference.
That said, its really hard to be comfortable with wet feet, and you'd be amazed how soaked your feet can get playing a round in the morning... with nothing more than grass wet from morning dew. Hence the popularity among disc golfers of decent hiking shoes/boots with some sort of waterproof membrane. They're designed for comfort on long walks in varied and rugged terrain and can keep your feet dry. Sounds a lot like playing disc golf to me.
But any ratty old pair of sneakers will work if the course is dry.