Of course they do. Anything containing carbohydrates will burn. For that matter, almost anything organic will burn.

Grapes have a very high water content, so they may require a little bit longer to ignite than something dryer, like wood. But given sufficient heat and/or time, they absolutely will burn. Once the water has evaporated out, the sugars in them will burn quickly and energetically.

In fact, if you pop some grapes in the microwave, you can make them burn spectacularly - they'll spontaneously create visible plasma.