“Never begin a sentence with but.” So my college freshmen tell me. They also tell me that people in the Middle Ages thought the earth was flat (everybody knew it was round), that women in the Middle Ages were no better than cattle (they had more freedom than they would enjoy until the twentieth century), that people in the Middle Ages were morose and grim (they were boisterous partiers who loved color), that they were morbidly fascinated with demons (they portrayed demons as ridiculous stooges), and they were oppressed by their kings (most of the kings were weak).
Again I open my Bible at random: “But I say unto you, I will not drink henceforth of this fruit of the vine, until that day when I drink it new with you in my Father’s kingdom.” And again: “But woe unto you, Pharisees!” And again: “But I know him, for I am from him, and he hath sent me.” If it is good enough for Almighty God, it is good enough for a freshman, certainly.
There never was such a rule in English grammar. Nor was there ever such a rule in classical Greek, or in Latin. It is the quick and natural way to begin an adversative sentence, one that shifts direction from the previous, or contradicts it, or backs away. Unless you have a particular reason for preferring the slower comma, however, comma, not only may you begin a sentence with but: you really should do it. I tell my students this all the time. But they are slow to turn from their old ways, the reprobates.