You are boiling potatoes on a gas stove and your friend suggests turning up the heat to cook them faster. Will this idea work?

If the water is already boiling then no. It will make no difference.
The boiling point of a liquid is the temperature at which vapour pressure of the liquid is the same as the pressure of the environment around the liquid and when the liquid changes state into vapour or gas phase. Water changes into steam.
Liquids cannot exist at temperatures above the boiling point unless changes are made to the external pressure conditions. Therefore in a standard cooking pan on a stove the highest temperature that water can achieve is 100 degrees C.
Turning up the heat will simply provide more energy but it will not make the water any hotter. However:
a) If the water has not yet boiled then turning up the heat will provide more energy allowing the water to reach boiling point faster.
b) If you replaced the pan with a pressure cooker then you can change the external pressure around the water and possibly achieve higher temperatures at the boiling point. That may cook the potatoes faster.
But on a normal cooking pan on a stove no the idea will not work..