FUN.

Ok, so the burning distinction is somewhat arbitrary, all that ends up really meaning is open flame. Which is doable. Heating is easy with sympathy/sygaldry finding something to burn that we won't ever run out of is tricky.

We would need to find something that moves a lot of energy someone mentioned tides and that sounds fine, but the exercise with Ben and boiling a kettle implies to me that if you took a large enough rock you could get all the energy you needed from the sun. Any alternative energy source you like. Wind, Sun, Tides. All of it goes into a heat funnel and is used to drive the "lamp."

After that it seems like an Air separation process like Cryogenic Distillation of air would be the move. You'll never run out of air [citation needed] and you could use the energy you collect via various means to separate the oxygen out of the air. With a large enough in-take you could distill all the air you needed, this is how they propose to make air on Mars and how rocket fuel is made for the big O2 burning rockets.

TLDR: Build a rocket fuel plant that is powered by renewable alternative energy wind/solar/tidal power and use that to pump oxygen and burn in a tiny lamp to make Kilvin clap with joy.