How to detect when an LED has burned out?
@BrianDrummond's suggestion is best. Find a range for the Vf at your current and (and over the applicable junction temperature range) apply a window comparator to the voltage. We use exactly that method to detect failure in an IR LED (or connections to it) since the obvious "visual" indication is not available. 

I've seen partial failures in some LEDs that result in dim light and a substantial change in Vf. Maybe something to do with the heterojunction structure or just a partial short. 

Window comparator from this website. Open-collector (or open-drain) comparators such as LM339 should be used, or diodes added. Note that quad comparator pinout is different from quad op-amp pinout.