TL;DR: This appears to have originated in 2013 when a 3-year-old Chinese girl died shortly after consuming persimmons and milk---her doctor was unable to eliminate persimmons as contributing to her death. Since then, multiple doctors have pointed to a lack of evidence that co-consumption of persimmons and dairy is poisonous, and called the claim a "rumor".
Excess consumption of (especially unripe) persimmons is understood to cause gastrointestinal stones. Co-consumption of persimmons and milk is not believed to be poisonous, but may cause indigestion so is not recommended.
Can eating persimmons harm you?
The medical keywords here are bezoar (and diospyrobezoar) and gastrolithiasis, or in lay parlance, stones in your digestive system. If you search for these keywords on PubMed (e.g.
persimmon bezoars) you'll find relevant case studies. Eating persimmon (among other foods) are believed to cause such stones:
The skin of unripe persimmons contains high concentrations of the persimmon tannin. Upon reaction with stomach acid, persimmon tannin is believed to polymerize and form a conglomerate in which cellulose, hemicelluloses, and various proteins are accumulated.
Iwamuro et al., Review of the diagnosis and management of gastrointestinal bezoars, World J Gastrointest Endosc., 2020.
The literature generally emphasizes unripe persimmons, since tannin "disappears" during ripening, and further notes their consumption in large quantities (e.g. "regular persimmon intake"; "a history of persimmon intake").
These stones are more common in places like China (noting that China produces the most persimmons), where the disease is called 胃柿石症 = "stomach persimmon stone disease":
Because Chinese individuals have a habit of eating persimmons, bezoars containing persimmons (especially giant bezoars) are common in China...
Huang, Cheng, Wei, Giant gastric bezoar removal from the stomach using combined dual knife–electric snare treatment: a case report, J Int Med Res., 2020.
The video at the top of the 胃柿石症 page is a news report where someone ate 6 persimmons in one sitting, and a month later had two "stomach persimmon stones" removed.
So how is milk relevant?
If we search in Chinese, we find a 2013 article where a 3-year-old girl who ate persimmons, and shortly after went to hospital and passed away:
[Google Translate]: A persimmon can kill people, is it true or not? What is the reason for this? Our reporter then found the hospital where the child had an accident. This is Dr. Li, the Deputy Director of Surgery who was admitted. According to her, the child arrived already in a coma. The situation is very serious. Unconscious, weak pulse, after diagnosis, the doctor believes that the child urgently needs surgery. Then we had an urgent physical examination, and the liver and spleen moved down.
[Google Translate]: After a preliminary diagnosis, Dr. Li believed that the child was suffering from gastric perforation and flatulence due to acute gastric dilatation, which caused the displacement of various organs in the body, resulting in various symptoms of acidosis. So, why does a child have a sudden stomach perforation, and why does it cause acidosis?
[Google Translate]: The doctor found a lot of persimmon peels in the child's stomach. Could it be said that this is the real cause of the child's stomach perforation? According to Dr. Li, there are many reasons for a child’s stomach perforation. On the one hand, the child’s stomach wall is weak, and there are signs of perforation long ago. On the other hand, it is not ruled out that it is related to eating persimmons.
3岁女童吃柿子胃穿孔死亡 柿子克牛奶螃蟹菠菜, 青岛全接触, 2013
So the doctor was unable to rule out persimmons as a cause of the girl's death. Beyond this, we have:
[Google Translate]: The expert in the video (Li Xu, Children’s Hospital Affiliated to the Capital Academy of Pediatrics) said, “The client ate two persimmons, bananas and drank a lot of milk on the day of admission. There was a large quantity of murky brownish-yellow liquid within chyme in her abdominal cavity, and a lot of persimmon skin.” The doctor also said, “There are many reasons for a child’s stomach perforation, and the child’s gastric wall is weak, and there are early signs of perforation.”
刘璟, 吃完柿子喝酸奶中毒？谣传！, 中国中医 (China Chinese medicine), 2015.
The above article flatly calls the claim (eating persimmons and drinking milk is dangerous) a rumor (谣传).
A Taiwanese site goes quote various experts (I omit all their credentials):
[Google Translate]: Jian Yuhua said: "The food combination of persimmons and yogurt or bananas is not toxic. ..."
[Google Translate]: Hong Ruisheng ... "There is currently no medical evidence that eating persimmons with yogurt will cause poisoning. ..."
[Google Translate]: Ye Ruiru ... "...but when you drink yogurt after eating persimmons, the tannins will meet the stomach acid and the protein of yogurt, and there will be some agglomeration and coagulation, causing indigestion."
The above article notes that the rumor evolved from persimmons (柿子) to pomelo (柚子). In regards to the deceased girl, they report she actually consumed persimmons and milk (not yoghurt).
From another debunking site we have:
There is a lack of scientific evidence to suggest that persimmons eaten in combination with yogurt or banana is poisonous.
Sim Ding En, Fake News Alert: Dietician Debunks Persimmon Myth, Gives Nutrition Tips, 2020, quoting dietician Heng Mei Shan.
So you can find doctors recommending not eating persimmons and dairy together, but not because they expect you to get poisoned.
Finally, searching PubMed for
tannin+dairy certainly doesn't turn up anything that indicates someone could be harmed by this combination in particular. There's also a logical inconsistencies in that coffee and tea contain tannin and a regularly drunk with milk worldwide, and there are many recipes containing persimmon and milk such as this one which has almost no other ingredients.