A small current comes out of the ADJ pin, about 100µA according to TI datasheet, so, the current in the resistor is lower than in the Being the ADJ terminal is the one at lower potential it has to put some current out, no way it goes in.
The LM317 is a voltage regulator, who is doing everything it can to put 1.25V between the ADJ and the out, that way, in conventional use, you have a constant current source in the resistor going between those two pins and the second resistor has that current (plus the Iadj) Then, the voltage across the resistor from ADJ to ground is the final voltage reference at a a constant current. You do need a minimum load for it to regulate, the nice thing is as you have effectively a current source there loading it, you could configure it to chew the minimum current so it does regulate when the complete circuit is unloaded. If you know you will be loaded there is no need for it and you can go for higher value resistors. That's why some applications call for the 220? resistor (when it's going to be loaded) and sometimes go for lower, 120? which already provide the minimum load.
Now, for current source you are replacing the resistor between ADJ and ground with your load, minimum load requirements remain the same so you are limited on how low your current could be used. >10mA would be the recommended value, so, don't go over 125? for your setting resistor. Anyway for that ranges makes much more sense to just use a jfet which will do the job quite good, if a single jfet doesn't cut the requirements a second one could be added making the output impedance of the source much higher. There are some devices designed to work just like that, for quite low currents J500 to 511 have fixed currents. Here is something to read if you are into that kinda stuff
http://www.vishay.com/docs/70596/70596.pdfJS