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:
一个柿子就能令人致死，是真是假？这其中又有什么原因呢？我们的记者随后找到了孩子出事的这家医院。这位就是当初接诊的外科副主任李医生，据她说，孩子送来的时候，已经是陷入了昏迷状态。情况非常的严重。神志不清，脉搏微弱，经过诊断，医生认为，孩子急需要手术治疗。然后我们紧急查体，肝脾下移。
[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.
经过初步的诊断，李医生认为，孩子是由于急性胃扩张导致了胃穿孔，胀气，使身体内的各个脏器移位，从而出现了酸中毒的种种症状。那么，孩子为什么会突然的胃穿孔，又为什么会引起酸中毒呢？
[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?
医生在孩子的肚子里找到了大量的柿子皮，难道说，这就是导致孩子胃穿孔的真正原因吗？听李医生说，导致孩子胃穿孔的原因有很多，一方面孩子的胃壁肌层薄弱，早就有穿孔的迹象。而另一方面，也不排除和吃柿子有关。
[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:
视频中的专家（首都儿科研究院附属儿童医院李旭）表示，“当事人入院当天吃了两个柿子，还有香蕉以及喝了大量的牛奶，孩子腹腔中有大量棕黄色的食糜中的浑浊液体，还有好多柿子皮。”医生也表示，“导致孩子胃穿孔的原因有很多，而且孩子的胃壁肌层薄弱，早就有穿孔的迹象。”
[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):
簡鈺樺表示：「柿子與優酪乳或香蕉的食物組合是不會產生毒性的。……」
[Google Translate]: Jian Yuhua said: "The food combination of persimmons and yogurt or bananas is not toxic. ..."
洪睿勝……「醫學上目前沒有證據指出吃柿子配優酪乳會中毒。柿子跟破布子因為富含鞣酸，……」
[Google Translate]: Hong Ruisheng ... "There is currently no medical evidence that eating persimmons with yogurt will cause poisoning. ..."
葉睿儒……「……但吃完柿子再喝優酪乳時，單寧酸碰到胃酸、優酪乳的蛋白質，會有一些結塊、凝固的情況，造成消化不良。……」
[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."
【錯誤】網傳「吃完柿子／柚子千萬别喝優酪乳，也不能吃香蕉，會中毒。有個孩子還沒到醫院就往生了」？, 2018
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 persimmon+milk,
persimmon+dairy,
tannin+milk,
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.
