With plastic lenses, like on light fixtures, sometimes the plastic turns brown from heat exposure to the hot bulbs.

Can this be removed with retro brite formula or hydrogen peroxide, or is it permanently discolored in this case?

At first, it is not just heat what causes it, it is the combination of heat, oxygen presence and the biggest contributor the intense high energy (mainly UV, but to some extend the shorter wavelength blue as well) photon irradiation.

What to use and whether it is even possible to "clean out" depends on the exact material.
For PolyCarbonate you may try some of the "refresh car headlights" procedures.
Try to google "RetroBright" receipes, but these are limited to only certain materials also (don't remember now if it was PVC and/or ABS or what), it is described there...

Correct me if I'm wrong Medved, but one thing to note with polycarbonate, especially automotive headlamps, is that the UV protection is only a surface coating. Polishing will remove the discoluration / crazing but also removes the coating. So unless the coating is restored after polishing the polycarbonate will discolour again very quickly.

Yes, the coating should be redone after the polishing. So indeed, it will never be as new, but on the other hand if it is already darkened to the extend of unusability and a new lens is an unobtanium, there is not much else you can do.
And also don't forget unlike the car hearlight, the HID lenses are exposed by the heat and UV from the inside, so the polishing and UV treatment should be applied from the inside...