I am not aware of any specific insect that cuts the flower bud of hibiscus at the base of the bud but there are problems with the buds that cause them to fall from the stem before opening.

There are two culprits that cause buds to drop off early, the recently introduced flower bud midge and a group of caterpillar larvae that feed inside the buds. These can be prevented by spraying the newly developing buds with a product such as Ortho Rose and Flower Insect Killer. Always read and follow the directions on the pesticide container for your safety and consistent control of the pests.