As an editor, I just had to weigh in on this subject.
As several people have already pointed out, the word “but” is used as a conjunction. Therefore, strictly speaking, its job is to link two independent clauses, not to begin a new sentence.
Obviously, in practical terms, when you begin a sentence with “but,” your reader can presumably figure out that you are now advancing a new idea that is somewhat contrary to the the idea in the previous sentence. In other words, you are not committing a really harmful error, which is one that leads to confusion on the part of the reader.
But–you should realize that you are breaking a rule. Therefore, you should only do it sparingly, as a means of emphasis (as I just did in this paragraph).
If your editor is objecting, then you should go back and see whether you are overusing this sentence construction. There are several other words that would serve your need as well (however, unfortunately, nonetheless, nevertheless). Not only would they add variety; they are also more flexibil, because you can place them at different points in the sentence.
For example: “I wanted to go for a walk, but it was raining.” It would be foolish to break this up into two sentence like this: “I wanted to go for a walk. But it was raining.” The second sentence just isn’t important enough to deserve breaking the rule. Why the extra emphasis?
However, if you’ve just used two or three sentence with the word “but,” and you’re looking to break the monotony, you could try:
“I wanted to go for a walk. However, it was, however, raining” or “It was raining, however” or “It was, however, raining.”
This is one of those rules like not using parallel fifths in music: if you do it, it doesn’t really hurt, but if you avoid it, you may find yourself coming up with something much better and more interesting.