Is it ever okay to start a sentence with the word me?

I get scolded a lot for saying "Me and Sal." What is actually wrong with that? While I have your attention, what about "I ain't got no pencil?" I am a proud product of the Parma City Schools.

If me is the object of a sentence (or part of the object), and the sentence is constructed appropriately, then you can start it with me.

See When do I use “I” instead of “me?” for a related discussion of I versus me.

The following sentence starts with me:

✔ Me and Sal: tell us what you want.

The object of the verb (telling) is us, which is composed of me and Sal.

For example:

✘ Tell I what you want.
✔ Tell me what you want.

It's the same (grammatically) as this:

✔ The ball: kick it here.

The object of the verb (kicking) is it, which is the ball.

The only thing that normally prevents this type of sentence from being used is that it's unusual. But that's a matter of style, not grammar.

There could be other types of sentences that could also start with me.

If you are being scolded for using me and Sal, then it's likely you're using an ungrammatical construction.

For instance:

✘ Me went to the store.
✘ Me and Sal went to the store.
✘ Sal and me went to the store.

✔ I went to the store.
✔ Sal and I went to the store.
？ I and Sal went to the store.

The final version is actually grammatical (in terms of syntax), but it's not stylistically normal. Typically, we put I last. So, the final version could be considered "wrong" by many people. However, despite violating conventional style, it is, nonetheless, grammatical in terms of following the rules of syntax. And if somebody does consider it wrong, it's not because of the use of I specifically, but because of where it's located.