One possible use is with Slotted Optical Switches (e.g. OPB961) for position measurement.
The position of a flag at which a slotted optical switch triggers depends on the intensity of the light from its LED. Which is a function of the current flow in the LED. If the current is not constant the light intensity isn't and thus the switch triggers inconsistently with respect to position.
As the temperature changes the forward voltage of the LED changes thus voltage regulation yields inconsistent position detection with regard to temperature. Using a regulated current source to drive the LED keeps the light intensity consistent (even with temperature) so the switch triggers consistently at a position of the flag.
Driving the LED with more than 20mA will cause it to be warm/hot and accelerate the aging process of the P-N junction, which will also change the light intensity.
One other side note: sometimes these Slotted Optical Switches are used in hazardous areas (e.g. explosive atmospheres, like dust from grain) so it is important that the current source can operate the IR LED through something called an Intrinsic Safety Barrier that is rated for the hazard and insures an arc or spark will not cause a problem (ignite the grain dust) should a failure occur. This basically means the LED string will be ground referenced.
Update: this is easy to do with a LM317, which has all the parts and a 1.25V reference pin...
http://www.onsemi.com/pub_link/Collateral/AND8109-D.PDF
