While it is never advisable to use the word "never" when it comes to English grammar rules, many grammarians still considered it unacceptable to start a sentence with and, but or because. In their opinion, doing so creates a sentence fragment, not a complete sentence. "And," "but" and "because" are used primarily to join two independent phrases together and create a relationship between them.
If you start a sentence with and, for example, it can be argued that you are actually creating one half of a more complex sentence. And people who are strict grammarians may object. The proper use of "and" calls for a conjoining of two separate ideas which are more or less equal in importance: "The coffee is in the pantry AND the eggs are in the refrigerator." There is an equal relationship between the coffee and the eggs, and the conjunction and establishes this. If you start a sentence with And, you are weakening that sentence unnecessarily.
The same grammatical rules apply when you start a sentence with But, but many people seem to feel more comfortable breaking with tradition here. The relationship between two contrary or opposite phrases could be established with the conjunction but: "I wanted to see a movie, BUT my wife wanted to read a book." The word "but" should not be used to start a sentence, because the relationship has not been established yet. But some people consider this rule to be a bit archaic.
There are, in fact, many examples in both literature and formal professional writing in which both And and But are used at the beginning of sentences. In many of these cases, the way in which these conjunctive words are used can add style and specific shades of meaning to the sentence. Although not all grammar experts agree that And and But should not be used at the start of a sentence, you may encounter resistance should you choose to use them in this way.
It is perfectly acceptable to start certain sentences with Because, as long as the sentence contains a cause-and-effect relationship: "Because the principal will not be in the building, the assembly has been postponed." This is a proper sentence, because there is a definite cause-and-effect implied. Some writers, however, may choose to use "Because" at the beginning of a sentence without such a relationship. Why would they do this? Because it works under the right circumstances.
When it comes to informal writing, it would be impossible to say a writer cannot start a sentence with and, but or because. As long as the message as a whole has been communicated effectively to the reader, the occasional sentence fragment posing as a complete thought shouldn't be completely discouraged. In more formal writing situations, however, a writer should rarely if ever start a sentence with and, but or because unless it is used to recreate the authentic dialogue of a quoted character. Otherwise, it is largely considered a sloppy practice among professional writers.