In Wheel of Time, Nynaeve was able to Heal both severing (stilling/gentling) and later the madness from the taint. Keeping to the philosophy that anything except death can be Healed, has there been an instance where a channeler who burnt through his/her ability to channel has been healed of it? This may have occured due to a ter'angreal, using too much of the One Power, or any other reason.
No, burning out can't be healed.
There are four relevant interviews in the WoT interview database which touch on this subject.
In Sep 2005 (item 33 here), Robert Jordan RAFOed the question:
Q: With regards to Healing stilling vs. being burned out. There are a lot of theories online that you can Heal stilling but you cannot Heal being burnt out. Is that true?
RJ: Read and Find Out.
In Nov 2009 (item 4 here), Brandon Sanderson said Jordan hadn't confirmed either way:
Q: ...anyway the question was if burned out channelers could be Healed as severed or stilled channelers could.
BS: His answer, paraphrased because I wasn't recording and I was laughing at what he wrote was: "Nynaeve thinks that there is nothing that cannot be Healed. She was right about Healing Stilling and she is on the right track." But he didn't confirm or deny or RAFO whether burned out channelers could or could not be Healed.
In Feb 2013 (item 20 here), Brandon Sanderson said burning out can't be Healed in the same way as stilling/gentling:
Q: Regarding burning out, is it Healable?
BS: Not in the same way that stilling is Healed. All it says in the notes is what I just said. And that Nynaeve would not be able to Heal Setalle. Being reborn would heal burning out, but transmigration would not. That's why even though Lanfear shifted bodies, she was still reduced.
In Apr 2013 (item 14 here), Maria Simons said burning out cannot be Healed, full stop:
Q: Can someone burned out be Healed, be it similar to the severing healing process or not?
MS: No, someone who is burned out will stay that way. It cannot be healed.
This makes sense because burning out is in a way more severe than being stilled or gentled: someone who burns out can't even sense the Source any more, whereas stilled/gentled people can, tantalisingly. From World of WoT:
The severing of a man from the True Source is now known as 'gentling'. He can still sense the Power, but is unable to touch saidin in any way. [...] The stilled woman, like the man who has been gentled, is cut off from the True Source, always tantalized by the sense of saidar, yet unable to touch or channel it.
The woman who is burned out can neither channel nor sense the Power.
So in this way it makes sense that stilling/gentling can be Healed while burning out can't.
This still does not follow the logic offered in the first premise. It has been maintained in the books that anything is healable except death. I know that there is a difference between stilled/gentled channelers. However, we know that there is an actual connection that Rand saw connecting the Chosen to the Dark one and this connection is similar to the connection to the one power. If there is a flaw in my argument then please point it out to me because if it's vulnerable to criticism then death is not the only thing not healable and thus the WoT canon concerning healing is fallacious.
To clarify I never implied that healing could not be accomplished by either sex. The canon confirms this. I was just going with the knowledge we the readers have as to what is necessary to healer a channeler properly. This string is concerning burned-out individuals and the evidence the canon has proven. We can't say for a certainty that burned-out channelers are a lost cause. There are areas in which a channeler cannot sense the One Power (Steddings, and Far Madding.)
The premise of anything but death is healable in the WoT books. Stilling was once believed not able to be healed. However, it has and requires the opposite sex to heal males and females. As a result, what do you think of the following logical argument below:
Anything can be healed except death Stilled channelers can be healed Thus, stilled and burned out channelers can be healed
Perhaps burned-out individuals can be healed using a circle of a man and woman (the man has to lead to healing a burned-out woman) or a circle where the woman can guide (healing a burned-out man). In any case, this could be the answer to burned-out healing channellers, just an idea I've been pondering.