Can you boil water in a Styrofoam cup over an open flame?

No.

You can boil water in a paper cup over a flame although you'll have more success if you have a steel grid in between so that the paper is heated evenly. The temperature at which paper burns is lower than that at which water boils so that the water keeps the paper too cool to catch light.

If you try the same with a styrofoam cup, it will melt. Because styrofoam is a good insulator, the water can't keep it cool enough to prevent it melting.