Can a wine be so old it's unsafe to drink?
My kind friend, you either are sipping too slowly or have inherited a 650-year-old cellar infested with the Black Death.
I'm no doctor (as my meagre retirement-savings account will attest), but my experience would suggest that wine's ill effects have to do with drinking too much or too quickly rather than too slowly. Fermentation is a liquid's first line of defence against decay, a natural cleansing process that enables juice and food to last longer than it would fresh. Acidity, sugar and, most vitally, alcohol also come into play, further extending shelf life.
I should point out a verbal distinction. Most wines tend to go "off" pretty quickly, in a matter of several years. But "off" in this context generally means unpleasant tasting, not unsafe as in rotten milk. (Only the best wines will improve with decades in the cellar.)
Health-wise, there is in almost all cases little to worry about even with the oldest wines you'll come across. I've had Bordeaux going back to the U.S. Civil War period of the 1860s and it was perfectly sound from a health perspective, though it was too far gone hedonistically to inspire me to bid on 150-year-old wine at auction.
The Flavour Principle, by Lucy Waverman and Beppi Crosariol, was named best Canadian Food & Drinks Book in the 2014 Gourmand World Cookbook Awards. Published by HarperCollins.