The resistance required to vary the current between 10mA and the full 1.5A is not linear (graph1). The variation of resistance required at high currents is minute. However, the power dissipated by the resistor varies linearly with the current sunk by the LM317. This presents a problem, as most 9, 16 and 24mm rotary potentiometers are limited to a power of 0.5W max, which would cap the current drawn by the circuit at ~400mA before catching fire. Horizontal and vertical trimpots are worse, as they can only dissipate 0.1W, which equates roughly to ~75mA. Furthermore, you would have great difficulty varying the current drawn as almost 80% of the current range is adjusted using the last 1-2 degrees of rotation.
There are several solutions to this problem:
Use a 15-25 revolution ‘multi-turn’ precision trimmer potentiometer, ‘3006P Horizontal’ style. Unlike normal potentiometers which typically start at 1 kiloohm, they are available in 50 ohm values and each 360 degree revolution equates to 2 ohms, allowing a much finer adjustment of resistance. They are available in power ratings of 0.75W max, so this limits the current draw to 600mA. This may be acceptable if you do not require a high current draw.
Use a high powered wirewound potentiometer. These are only available in very low resistance values, typically 5 ohms, but can handle up to 15W of power. With a total rotation angle of 300*, this allows for an adjustment of 0.017 ohms/degree. The low resistance (5 ohm) does mean that the minimum current draw is 250mA, but you are able to draw the full 1.5A+ without burning out the resistor.
If you are lucky enough to own or come across precision laboratory rheostats, they can also be used to similar effect.