The hypothesis that Stalin was poisoned has been supported by both Stalinists and anti-Stalinists. Among the neo-Stalinists in Russia it's almost a dogma that Stalin was killed (probably by Khrushchev). Those anti-Stalinists who accept this usually blame Beria (who is usually revered by neo-Stalinists, so they cannot blame him).

Now, this hypothesis is not as outlandish as Hitler escaping Germany. It's more on the JFK conspiracy level.

Oleg Khlevniuk in Stalin. New biography of a dictator, 2015, pp. 191ff tells at length about Stalin's poor health. P. 196:

He continued to suffer from stomach pain and intestinal disturbances, accompanied by fever, throat problems, colds, and influenza. His atherosclerosis was progressing. Despite scattered attempts to do so, he was by now simply incapable of changing his sedentary lifestyle. The copious fare served at his frequent late-night dinner gatherings was surely not good for him. According to Milovan Djilas, who visited Stalin’s dacha several times in the 1940s, “The selection of food and drink was huge, with an emphasis on meat dishes and hard liquor.”

[...]

In 1952, Stalin did not travel south. Even though he remained in Moscow, he visited his Kremlin office only fifty times, an average of less than once a week. On 21 December 1952, for his seventy-third birthday, his daughter Svetlana made her final visit to her father’s dacha. “I was worried at how badly he looked,” she recalled. “He must have felt his illness coming on. Maybe he was aware of some hypertension, for he’d suddenly given up smoking and was very pleased with himself. . . . He’d been smoking for fifty or sixty years.” By this time his atherosclerosis was well advanced. The autopsy performed two and a half months later showed that damage to the arteries had greatly impeded blood flow to the brain.

He also infamously did not trust doctors anymore. Albeit we don't have any hard evidence, pro or contra, about the possible doctor care in the last months. But the hesitancy to call a doctor on the part of his underlings could have been caused by this paranoia (if not malice...) and thus possibly contributed to his death. Pp. 189-190:

The renowned Soviet cardiologist Aleksandr Miasnikov, one of the medical experts summoned to attend Stalin, gives a detailed description of the visit in his memoirs. “The diagnosis,” he wrote, “was clear to us, thank God: hemorrhage in the left cerebral hemisphere of the brain caused by hypertonia and atherosclerosis.” The doctors gave Stalin generous doses of various stimulants but without any real hope of preventing death. From a medical perspective, his condition was no mystery. An autopsy confirmed the initial diagnosis, revealing a large cerebral hemorrhage and severe damage to the cerebral arteries due to atherosclerosis. 190 patient number 1 Stalin had been a sickly old man. He would have turned seventy-five later that year.

There's no credible evidence that Stalin's death was anything but natural. The only piece "pro" is Molotov's claim that Beria told him that he "removed him":

- They say he was killed by Beria himself?

- Why Beria? It could have been a security guard or a doctor, - Molotov answered. - When he was dying, there were moments when he regained consciousness. It was - he was writhing, there were different moments like that. It seemed that he was beginning to recover. That's when Beria held on to Stalin! Ooh! He was ready... I do not exclude that he had a hand in his death. Judging by what he said to me and what I myself sensed.... While on the rostrum of the Mausoleum with him on May 1, 1953, he did drop hints.... Apparently he wanted to evoke my sympathy. He said, “I removed him!”. As if he had benefited me. Of course he wanted to ingratiate himself with me: “I saved all of you!” Khrushchev would scarcely have had a hand in it. He might have been suspicious of what had gone on. Or possibly... They had been close by after all. Malenkov knows more, much more, much more

That's not only vague and uncertain (Molotov didn't mention anything like that during the anti-Beria TsK Plenum in July of 1953), Beria also wasn't accused of anything like that during the investigation and trial - and this would have been a perfect accusation at the time (Stalin's cult of personality continued to exist for some time and during the Plenum Beria was accused of anti-Stalin remarks).

In any case, the natural cause of death - cerebral hemorrhage - is simply the most parsimonious explanation. It was also confirmed by the cardiologist Aleksandr Nedostup from the Moscow Medicine Academy of Sechenov in 2003, when he was officially tasked with checking the medical records of Stalin's last days:

- Did you conduct an examination of Stalin's last medical case history?

- Yes, at the request of the Central State Archives. In 2003, by the 50th anniversary of Stalin's death, some publication was being prepared for printing, but it never came out, there was not enough money. Still there are some books, studies are conducted to find out the true cause of Stalin's death - whether it was murder or not.

After studying the documents, I came to an unambiguous conclusion. I held this medical history in my hands, which cannot be called a medical history in the modern sense. These were bound sheets of paper with a binder or paperclip, which were recorded every 15, 20 or 30 minutes. Sisters on duty, doctors on duty, were writing, and the decisions of the councils were made there. We could see how all these people were worried, how they repeated the same thing, how their hands trembled, how their handwriting changed. Theoretically, of course, you can fake any medical history, but you cannot fake it specifically this way. Indeed, it was a very serious stroke, pressure that did not decrease, because there was no way to reduce it, there was no medicine.

One would have to have a good reason to even begin to entertain the poisoning thought seriously. Authors like Brent and Naumov, who in their (scholarly quite poor) book Stalin's last crime propose the use of warfarin, don't cite any such good reason. So it's not something that can - or should - be discussed seriously.