The LM317 is thinks as a voltage mode device. It has an internal reference voltage, and manipulates the output in whatever way it can such that the output is greater than the adjust pin by exactly that reverence voltage amount. Because a resistor has a directly proportional relationship between voltage and current, the 317's attitude about voltage has the side effect of causing a constant current through the resistor. But that's all it is, a beneficial side effect. The 317 is regulating voltage and only voltage. Once you see the world from the 317's point of view, you can better determine how it will react to other impedance devices.
With sufficient sensing and feedback control circuits, a bipolar transistor can emulate a constant resistance in one direction of current flow. This is how some electronic loads (power supply test equipment) work. It takes a little less work with a FET because its method of conducting current is closer to that of a resistor in the first place. Still, it won't work without some control circuitry. Same for using two diodes in a diode switch configuration.
So to answer your original question, yes it is possible. But it probably is not practical unless there are some unusual application requirements.
ak