What does a Time Lord need to regenerate?
As far as we know, a dying Time Lord only needs one, at most two things in order to regenerate:
Another regeneration to use. It has been previously established that, under ordinary circumstances, a Time Lord can only regenerate twelve times. Once that limit is used up, there's nothing they can do. We've seen this in action with the Master who, at the very end of his regenerative cycle, is forced to use other means of extending his life
The desire to regenerate. In "The Sound of Drums", we saw that a Time Lord (the Master, specifically) could refuse to regenerate:
Doctor: You're not dying. Don't be stupid. It's only a bullet. Just regenerate.
Doctor: One little bullet. Come on.
Master: I guess you don't know me so well. I refuse.
Doctor Who Series 3 Episode 13: "The Last of the Time Lords"
Although the Master is an unreliable source, the Doctor himself confirmed in "Twice Upon a Time" that it's possible (though apparently difficult) to simply refuse to regenerate, the result being permanent death:
Twelfth Doctor: Why are you trying not to regenerate?
First Doctor: I have the courage and the right to live and die as myself.
Twelfth Doctor: We have a choice: either we change and go on, or we die as we are.
Doctor Who "Twice Upon a Time"
So what was different in "Heaven Sent"?
The Doctor himself gives us a hint in the episode (emphasis mine):
Doctor: People always get it wrong with Time Lords. We take forever to die. Even if we're too injured to regenerate, every cell in our bodies keeps trying.
Doctor Who Series 9 Episode 11: "Heaven Sent"
Whatever the Veil does to him, it seems to bypass the regenerative process. We know this is possible, and have seen it done (or nearly done) on a couple of occasions:
- We know that the Time Lords can execute their own; they do it to Morbius, and attempt to do it to the Doctor
River Song warns the Tenth Doctor of this in "Forest of the Dead":
Doctor: Easy! We beam all the people out of the data core. The computer will reset and stop the countdown. Difficult. Charlotte doesn't have enough memory space left to make the transfer. Easy! I'll hook myself up to the computer. She can borrow my memory space.
River: Difficult. It'll kill you stone dead.
Doctor: Yeah, it's easy to criticise.
River: It'll burn out both your hearts and don't think you'll regenerate.
Doctor Who Series 4 Episode 9: "Forest of the Dead"
Since the Veil appears to be electrically-based, this seems like a particularly plausible explanation
We know that the Doctor can die if he's killed mid-regeneration:
Wilf: I thought, when I saw you before, you said your people could change, like, your whole body.
Doctor: I can still die. If I'm killed before regeneration, then I'm dead.
Doctor Who "The End of Time Part 1"
River infects the Doctor with the poison of the Judas tree, which appears to have this effect1:
Holo-Amelia: Your system has been contaminated by the poison of the Judas tree. You will be dead in thirty two minutes.
Doctor: Okay. So, basically better regenerate, that's what you're saying.
Holo-Amelia: Regeneration disabled. You will be dead in thirty two minutes.
Doctor Who Series 6 Episode 8: "Let's Kill Hitler"
Granted, at this point the Doctor is on his last regeneration anyway. However, it seems unlikely that River - an assassin raised from birth with the express purpose of killing the Doctor - would choose any means of killing him that didn't bypass regeneration. It is unclear, though
1 Nod to tilley31 for reminding me of this case