Not much. Carbohydrates can be used by our bodies for energy. But so can fat and proteins. Any cells that contain the little organelles called mitochondria are able to use fatty acids (which come from fat) and ketone bodies (which are made from fatty acids and from the ketogenic amino acids in proteins*) for energy. In fact, some tissues prefer non-carbohydrate energy sources.
There are only a few tissues in the body that must use carbohydrates as their energy source--some parts of the brain, some kidney cells, red blood cells and the cornea of the eye. These tissues will use about 40 grams of carbohydrate per day, which is well within the margin that can be produced by gluconeogenesis in the liver.
*Corrected from the original version. Thanks to Hans, in the comments.