They actually do, but let's set Relativity aside for a moment. And the whole barycenter thing too. Here's the scenario:
Somewhere in the backwaters of a galaxy there is this huge planet. It is NOT a failed star. It could be considered a gas giant, except most of the gas is nitrogen, with traces of oxygen. The planet has a nucleus with a rocky surface covered in part by water. There's life on the planet, maybe even intelligent life. Orbiting this planet is a dwarf star. It is sufficiently close to the planet to provide energy and warmth, and not close enough to incinerate it. Its orbital speed is sufficient to warm the equatorial parts of the planet without singing them when it passes over them and to come back to them before they freeze over. Or maybe there are two stars of slightly different sizes, traveling along different orbits.
Is this at all possible?