It's not clear what you mean by "feed off" exactly.
Any radiation a star or black hole emits must come from the energy density inside it. For stars, the source of that energy is fusion in its core, and that energy is emitted as thermal radiation in the photosphere. Black holes are thought to emit Hawking radiation near the event horizon, though at much lower intensity than the light coming from stars.
Does that answer your question?
Not in a perpetual motion sense, no. Stars feed off their own energy in the sense that they're converting their gravitational energy into heat and pressure, but that process will eventually use up all their fuel.
All the time, that’s how stars work.
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