The analogue input from the potentiometer (Manual 3 page 27) does not have input protection (via a 1kOhm or similar) shown. If the pot wiper is moved to either extremity then the PICAXE input is either at ~5Vdc or ~0Vdc.
But, as said, the resistors are not techncially essential but can prevent prevent mishaps causing PICAXE damage.
A situation where a pin is set to an output (and this may be a simple programming error) and say high then a circuit such as the potentiometer circuit on page 27 is attached and set to the minimim value (~0Vdc) and the PICAXE output is shorted causing overload of the output and failure of the PICAXE.