The resistor and the trimmer form a voltage divider (*) that provides a voltage to the ADJ pin, and the regulator will tweak the output voltage to keep ADJ at 1.25 V. The pin itself only consumes 50-100 uA according to the datasheet, so the main load will be through the resistor & trimmer to ground. If you go too low, you may damage the trimmer, though – e.g. with 100 ohm the current is 1.25/100=12.5 mA which is a bit too much for a 10k trimmer (see here).
*) Well, in Sam’s sketch it’s not a divider. What was the idea behind not connecting the pot to ground? Seems to me that would turn it into a current regulator, not a voltage regulator, with the regulator trying to keep the current through resistor + trimmer high enough to drop 1.25 V, which doesn’t quite work since 1.25 V/50 uA = 25k. Are we sure it’s actually regulating when you’re seeing a 1 V drop, or could that be the Vce drop of the regulator’s power transistor?