According to MedlinePlus, milk-alkali syndrome results from the presence of excess calcium in the body and can lead to a person losing his kidney function. Too much calcium in the body shifts the body's acid/base balance toward alkaline. Vitamin D toxicity is another possible result from drinking too much milk.
The For Dummies website states that vitamin D toxicity is a possible consequence of poor production standards of milk. A local dairy in Massachusetts added quantities of vitamin D far exceeding the typical supplementary standard to its milk, causing 20 heavy milk drinkers to get vitamin D toxicity. Like milk-alkali syndrome, vitamin D toxicity is unlikely to occur through milk drinking alone. MedlinePlus states that the cause of milk-alkali syndrome is typically the ingestion of calcium supplements in very large quantities.