The Chicago Manual of Style says it all:
“There is a widespread belief—one with no historical or grammatical foundation—that it is an error to begin a sentence with a conjunction such as and, but, or so. In fact, a substantial percentage (often as many as 10 percent) of the sentences in first-rate writing begin with conjunctions. It has been so for centuries, and even the most conservative grammarians have followed this practice. Charles Allen Lloyd’s words from 1938 fairly sum up the situation as it stands even today:
Next to the groundless notion that it is incorrect to end an English sentence with a preposition, perhaps the most widespread of the many false beliefs about the use of our language is the equally groundless notion that it is incorrect to begin one with “but” or “and.” As in the case of the superstition about the prepositional ending, no textbook supports it, but apparently about half of our teachers of English go out of their way to handicap their pupils by inculcating it. One cannot help wondering whether those who teach such a monstrous doctrine ever read any English themselves.7
Still, but as an adversative conjunction can occasionally be unclear at the beginning of a sentence. Evaluate the contrasting force of the but in question, and see whether the needed word is really and; if and can be substituted, then but is almost certainly the wrong word. Consider this example: He went to school this morning. But he left his lunch box on the kitchen table. Between those sentences is an elliptical idea, since the two actions are in no way contradictory. What is implied is something like this: He went to school, intending to have lunch there, but he left his lunch behind. Because andwould have made sense in the passage as originally stated, but is not the right word—the idea for the contrastive but should be explicit. To sum up, then, but is a perfectly proper way to open a sentence, but only if the idea it introduces truly contrasts with what precedes. For that matter, but is often an effective way of introducing a paragraph that develops an idea contrary to the one preceding it.
7. Charles Allen Lloyd, We Who Speak English: And Our Ignorance of Our Mother Tongue(New York: Thomas Y. Crowell, 1938), 19.