Want to see a fire that's so determined that it will burn merrily away even though you rob it of oxygen? Then grab a big block of dry ice and some magnesium and prepare for horror! ...and cool visual effects.
Dry ice, which is carbon dioxide in its frozen form, is generally a great way to smother a fire. It's even a good way to smother a person, which is why you need to keep the windows open if you have it in a room or a car. The carbon dioxide will melt, expand, and displace the free oxygen in any given space. Whatever is in that space will get smothered.
That is, unless what's in that space is burning magnesium. Magnesium is "reactive," which is a polite way of saying that a chemical will do whatever it takes to make a reaction happen. Although nothing can burn without oxygen, carbon dioxide itself contains oxygen. It's good ol' CO2. Humans aren't capable of separating out that oxygen from the carbon that's holding on to it. Magnesium is. It will rip the oxygen right off the carbon dioxide molecule so that it can keep burning.
The most famous demonstration of this involves a block of dry ice and some magnesium shavings. Cut the dry ice in half and make a hollow in one half of the block. Put the magnesium shavings in and use a torch to set them on fire. Then settle the top half of the block back on top of the burning magnesium. The magnesium will use up its oxygen quickly, and should go out, but it doesn't. The block lights up, vibrates, and starts pouring out steam like something that needs an exorcism. The magnesium is taking the carbon dioxide apart so it can keep itself going.