The will burn for billions of years, it won't run out any time soon. This is sustainable because it will not deplete in the near future.
The harder question is if the materials that go into a solar cell are sustainable. They use primarily silicon, which is one of the most abundant materials on earth. But they can also use other things, like metals (copper, silver, gold), and some toxic chemicals (arsenic, cadmium) etc. The sustainability of these materials is questionable. These materials are recyclable, and it is thought 99% of a solar cell can be recycled. But the production of them does has some toxic site affects, which means we have to balance the 'clean' energy production of the cells with the bad by products of their manufacturing process.