Not as dangerous as some well-meaning friends who spread scary stories. But there is a small grain of truth to the story.
When potatoes were introduced into Europe in the second half of the sixteenth century, they were suspected of being either poisonous or aphrodisiac or, an intriguing thought, both. (What a way to die!)
Europeans tended to think the same of any exotic food from the New World, including tomatoes. (Their scarlet color no doubt helped to provoke the French into calling them pommes d’amour, or love apples.)
But we must let the suspicious Old Worlders off lightly, because both potatoes and tomatoes are indeed members of the same family, the nightshade, whose most infamous and deadly poisonous member is the belladonna plant.
I can’t help pointing out here that in Italian, bella donna means “sweetheart” or “good-looking woman.” Why was the plant so-named? Because it contains atropine, an alkaloid that dilates the pupils of the eyes. It was used (the story goes) by sixteenth-century Italian women as a cosmetic to simulate sexual arousal.
Fast-forward to the twenty-first century and your well-meaning friend.
The toxic alkaloid solanine, normally present in small amounts in potatoes, does build up in the eyes when they sprout. So eyes that are beginning to sprout should certainly be excised, and most especially if they have started to turn green.
But even then, the solanine doesn’t lie very deep, and an ordinary gouge with the paring knife will take care of it.