What would happen if I used a potentiometer to produce a 5V DC signal for an Arduino from a 9V battery?
The potentiometer is a resistor component. When you try to create a voltage reduction through a resistor the voltage drop is 100% dependent upon the amount of current flow through the resistor. A load like an Arduino is not a fixed current load so it is not possible to use a single resistor value to reduce the 9V to 5V suitable for proper operation of an Arduino.
You will need to investigate what a voltage regulator does and why it is applicable to the application.
Beware that if your 9V source is a typical 9V battery it is far from a suitable long term power source for an Arduino due to the relatively low capacity of such a battery.