Do star remnants actually burn? As we know, some stars can fuse heavy elements to produce carbon and oxygen after using up hydrogen if they are massive, and the remnant of stars are usually hot, my question is, if a star fuse to produce carbon and oxygen, does it also burn to produce carbon dioxide?
Stars don't "burn", they undergo nuclear reactions that do not involve atoms and chemistry at all. The temperatures in the interiors of stars, and certainly in the interiors of stellar remnants like C/O white dwarfs, are far too hot (millions of degrees) for electrons to bind to nuclei and far too hot for molecules to survive. The energy required to break up carbon dioxide is 5.5 eV, which is easily provided unless a gas is cooled well below 10,000K. 

The only parts of a star where chemistry can occur is in the outer atmospheres where temperatures can drop to thousands of degrees and where atoms and partially ionised atoms can exist. Here, yes, then it is possible for carbon atoms and oxygen atoms to interact, but that would mostly produce carbon monoxide. This molecule is produced and can survive between temperatures of about 1100K and 3500K in the atmospheres of cool M-dwarfs and brown dwarfs and the atmospheres of red giant stars.

These chemical reactions are utterly negligible in terms of their energetics compared with the fusion reactions that power a star.