(By stars I'm assuming you're implying stars like the Sun, which are a majority of the stars we see. @Dirk Bruere's answer about Black Dwarves is correct. )
No, I don't think they can. The primary process that 'fuels' stars is nuclear fusion. In the process of nuclear fusion, lighter elements fuse together, releasing a tremendous amount of energy (because the fusion product has greater binding energy than the reactants.) This, however, is only true until elements around Iron. Once Iron is obtained as a fusion product, it is not favorable to fuse Iron and its neighbors into heavier elements because the products obtained have lesser binding energy.
Stars exist in their observable state because of the intricate balance between the outward flux of radiation and particles from fusion, and the stars own gravity. Our sun, for instance, is in its Main Sequence stage, and is currently shining because of Hydrogen fusion. When the hydrogen in its core becomes significantly less, the drop in radiation pressure would cause it to collapse and constrict its core, which would cause helium to fuse. This cycle of expansion and contraction can only continue until certain elements, because some stars are just not heavy enough to provide energy for the fusion of heavier elements. (For example, the sun's fusion process would be favorable only till elements close to Carbon.)
For lower mass stars, when the end point of this cycle is reached, the outer layers of the star are shed off while the core stabilizes.
Core collapse takes places for very massive stars, because of reasons I mentioned in the first paragraph. (This, however is not the end of a star's life cycle. Read this: https://en.wikipedia.org/wiki/Stellar_evolution)
Summarizing, stars (in their shining form) can't live forever.