If I enclose an LED in "tin foil" so that the photons it gives off are reflected back to it, would the power consumed by the LED go down?
I guess the question is whether the LED will also act as a photodiode, i.e. whether incident light will excite electrons and therefore generate a current. If so, then reflect the diodes light back onto it will indeed reduce the current flowing in the diode by generating an opposing EMF.
This turns out to be surprisingly hard to Google, but Wikipedia suggests that LEDs will indeed act as photodiodes, and it cites a couple of papers to back up the claim.
However I'd guess that the photodiode efficiency will be low because LEDs are normally designed to have a direct band gap, which means excited electrons will recombine very fast and either just emit light again or dissipate their energy as heat. So while I'm not sure I would be prepared to testify under oath, my guess is that it will reduce the current but not by very much. The main effect will be to heat up the LED.