"Dynasty of One" by James White. You may have read it in his 1982 collection Futures Past. According to the ISFDB, a revised version was published as "Long Will Live the King" in Extro Science Fiction, April–May 1982, a British semi-prozine.
". . . The Just, the All-powerful, the All-knowing, His Celestial Majesty, Tate the First!"
He sat down.
Nobody moved anywhere. This was no ordinary function, where he granted audiences or issued the decrees which could alter the destiny of whole stellar systems. This was the time when he had to prove his fitness to rule, or die. In utter silence he pressed one of the two studs set in the arm of his throne, and tried to relax as golden bands of beautifully worked metal closed around his limbs, chest, and head, holding him rigid.
[. . . .]
When given before the age of forty and renewed every twenty years, the Immortality Treatment prevented the disease of senility and death from occurring in life based on the carbon series of compounds—which meant practically all forms of life. There were thousands of dogs, cats, and monkeys to prove that it worked. But in beings of higher intelligence—human or otherwise—it did not work at all, unless the being in question was mentally very, very tough.
The radiation which stimulated the regenerative centers produced other effects as well, some of them good, others quite fatally bad. The treatment increased the I.Q., and gave to the mind a perfect, eidetic memory. It also, for the few seconds duration of the treatment, so intensified the effects of what had come to be called the "area of conscience" that any being having sufficient intelligence to base his actions on a moral code had to take three seconds of the most frightful psychological torture ever known. He had to live with the cruel, debased, and utterly nauseous creature that was himself.
Many preferred to die rather than take three seconds of it. Most had no such choice—their life force was obliterated with the first, savage blast of self-knowledge.
This secondary effect of the treatment was experienced in a complete reliving of the past, with each incident diamond-sharp in visual, auditory, and tactile sense recall. But not only that. The mind was given a terrifying insight into the end results of that being's most trivial-seeming actions. Unthinking words or gestures made over the years and forgotten, when blown up by the triple stimulus of perfect memory, increased I.Q., and a hypersensitive "conscience" became lethal as a suicide's bullets. The mind just could not take such an overwhelming blast of self-guilt, even for three seconds, so it, and the body containing it, died.
Only one person had successfully undergone the Immortality Treatment.
Tate, though he had lived—with thirty-seven previous treatments—for seven hundred and sixty-eight years, still took only three seconds. And there was no blurring or telescoping of events. Each incident was complete, and though it occupied only microseconds of time, each bore its charge of guilt potential.