Can you rest and let sesamoiditis heal on its own?

Sometimes. "sesamoiditis" is a vague diagnosis encompassing many causes of sesamoid pain. If there is merely inflammation around the sesamoids, rest +/- immobilization can often resolve the problem.
Yes. When treating sesamoidiitis it is important to let the injured part of the foot heal. For severe cases of sesamoiditis immobilization in a cam walker ( walking boot is needed). For mild to moderate cases orthoitcs with a dancers pad is usually recommended. Consulting a podiatrist for this issue is always recommended because what you think might be sesamoiditis might be a fracture.