What happens when a stars core runs out of oxygen?

Unless a star ts very old and very massive, it will not consume oxygen. Stars are powered by nuclear fusion, which fuses hydrogen into helium. When a star runs out of hydrogen at its core it expands into a red giant and starts fusing hydrogen in a shell around the core. If the star is not massive enough to fuse helium, then it will shed its outer layers and leave a helium while dwarf where the core was. If it is massive enough, it will fuse helium into heavier elements up to oxygen. Most stars to not make it past this stage. More massive stars, though fuse elements such as carbon and oxygen into neon, magnesium, and sulfur.

If at any stage the star can fuse no more, the it sheds its outer layers and leaves behind a white dwarf.

The exception is if the star makes it as far as fusing iron. If that happens the core will stop producing energy. The core will collapse into either a black hole or a neutron star and the rest of the star's mass will be blown away in a massive explosion called a supernova.