Short fantasy written by a Brit in the 1960s or 1970s. Maybe by Brian Aldiss, but I am not sure.
A medical process makes the entire human race immortal. After a time a strange thing begins to happen - babies are born with no emotions or empathy; they have no souls.
The reason: there is only a limited number of human souls, which are reincarnated again and again and again. With all souls immortal, there are no new souls for newborns. The solution: humans must begin to die again.
I'm pretty sure this is John Brunner's "The Vitanuls".
It has an semi-immortality pill (not perfect and just rolling out):
"I think I follow you," the matron said at length. "I take it the anti-death pill is a success?" ... "You in the rich countries like America and Russia have been trying for years to find a broad-spectrum specific against aging, and I think--knowing your countries by hearsay--that there must have been a loud angry argument over who should benefit first."
It takes place in India, which gives them the setup to bring in reincarnation as a Chekov's Gun:
The matron gave him a chill stare. "I think that is uncivil of you. The teaching of Hinduism is that the soul is born again, throughout an eternal cycle, until it achieves one-ness with the All."
I can't provide direct page quotes for the ending (I was quoting from Google Books and they omit the next few pages), but a summary does indeed indicate that babies do start getting born with a curious lifelessness, and it is the lack of enough human souls that is the cause.
And there is this line:
"No, that's absurd. You can't mean to maintain that we've run short of human souls, as though they were stored in a cosmic warehouse and issued off the shelf every time a child is born!"
Could this be "The Vitanuls" by John Brunner?
"The Vitanuls" (1967) proceeds on the premise that the current population explosion, plus the introduction of a longevity drug, has used up the entire pool of human souls and that babies are being born mindless. An Indian holy man, once a noted obstetrician, chooses death in order to free his soul for another.
Voices for the Future: Essays on Major Science Fiction Writers, Volume 2
You can read an excerpt here
Is it Brian Aldiss's Hothouse?
I haven't read the entire plot summary, but here's an excerpt from the book (emphasis mine):
'You are still young,' said the Chief Captive. 'You have entered a fresh life. Where are your souls?'
Lily-yo and Flor looked at each other. In their flight from the wiltmilt they had heedlessly thrown down their souls. Haris had trampled on his. It was unthinkable!
'You see. You needed your souls no more. You are still young, and may be able to have babies. Some of those babies may be born with wings.'
The Captive with the boneless arms added, 'Some may be born wrong, as we are. Some may be born right.'
