A common misconception is that the consumption of oxygen by the flame in the container is a factor in the water rising. There may be a slight possibility that there would be a tiny rise in the water from the flame using up oxygen but it’s extremely small compared to the actual reason. Simply put, the water would rise imperceptibly at a steady rate as the oxygen were consumed. You likely saw the level rise almost all at once and pretty much after the flame went out.
At first, the flame heats the air inside the container and this hot air expands quickly. Some of the expanding air escapes from under the vase where you might have seen some bubbles. When the flame fades and goes out, the air in the container cools and cooler air contracts or takes up less space. That contraction creates a weak vacuum – or lower pressure – in the container. Where’s the higher pressure? Right! It’s outside the container pressing down on the water in the dish. The outside air pushes water into the container until the pressure is equalized inside and outside the container. The water stops rising when that pressure equalization is reached.