It's tempting to say that the water keeps the paper from reacting with oxygen, which is needed for the oxidization process of fire. But the paper doesn't lack oxygen, it just lacks heat. Although the fire is scorching it from one side, the water is cooling it from the other. It can't get hot enough to burst into flame.
Paper burns at around 500 degrees Fahrenheit (although book paper burns at a relatively balmy and famous 451). Water boils at a mere 212 degrees. When the flames heat the paper to above 212, the water nearby turns to vapor steams away, leaving cooler water to take its place. The boiling water can't get above 212 degrees, and the paper is thin enough that this water keeps its temperature from climbing high enough to combust. It's only when the water on the other side of the paper entirely boils away that the paper cup will burn away.