What kinds of shoes should I wear if I have a nail disease?

Breathable. Wear a shoe that breathes not one that creates heat and perspiration.
Properly fitting. The toe box should be large enough so that the nails don't hit the tip of the shoe or toe box. Weather permitting open toed are good as sunlight helps kill fungus. Also, allows the toes to breathe.
Open Toe Shoes. If your nail disease is fungal, then keeping the feet open to air so they can stay dry will help. Fungal organisms need moisture to thrive. If your feet sweat then you set the nails up for the environment that promotes the fungal nail disease.