The Sun’s heat comes from Thermonuclear fusion at its core: light elements such as (but not exclusively) 2 Hydrogen atoms turning into a Helium atom.
This process uses up mass, and generates a LOT of energy. The Sun IS using up its fuel, but fusion is so efficient that the Sun will continue to shine for billions of years.
The burning you are thinking of (that requires Oxygen) is simple chemical combustion. It releases much less energy than nuclear fusion. But it is also much easier to do - Thermonuclear fusion is very hard to initiate and sustain, and might never turn out to be a good source of power on Earth.
Hope this is helpful.
Make that 4 Hydrogen atoms, oops!