There are countless recitations of the common practice of tasting for diabetes, but I can't find any example of other urine taste tests despite many sources stating it was used for other diseases and that urology wheels[1] encoded tastes and smells in addition to appearances. Have you found any?
The closest I could find was mention of smelling urine to determine pregnancy, which apparently was/is rather reliable. The next best was mention of the surgeon John Hunter discussing the taste of semen, though the author "assumes... [Hunter] was tasting his own". (I wouldn't assume that considering his interests and research. Google Books texts of Hunter's publications contained page after page of discussion about the genital fluids of male cadavers, but after his brief mention of the taste of semen from a live donor he never quite goes where you're fearfully hoping! I didn't get there in my skimming--so much preamble for stuff everybody learns in Zoology and Physiology 101--but I presume he ended up reporting many important discoveries about male reproductive tissues and processes.)
I wish Google Books was as awesome today as it was 10 years ago. They expose only a tiny fraction of the material they once provided, even for pre-20th century material that couldn't possibly be under copyright. :(
[1] I wonder if they were inspiration, perhaps indirectly, for the wine aroma wheel created by the chemist Ann Noble at UC Davis.