I realize many verbs add the "be" to the beginning, which fundamentally changes the meaning, e.g., kommen and bekommen, or stehen and bestehen. However, it seems as though the "be" in belehren doesn't change the meaning much, if at all. So when should I use it?
I disagree with you that it doesn't change the meaning that much...
means to teach something to somebody.
in most cases means that you're correcting somebody who thinks he knows, but is not really correct.
One other use of belehren would be that you inform somebody of something.
So to sum up, with lehren you're kind of teaching or providing knowledge to somebody.
With belehren you're rather telling somebody something than you're teaching.
There is a grammatical difference and a difference in meaning...
Belehren takes a person as a direct object in accusative case. If you want to include whatever is being taught that is done using the preposition über or hinsichtlich
Belehren has a notion of completion... at least more than lehren has. After having been belehrt, I expect the person to know everything about the matter. Also, belehren is done with rather limited fields of knowledge. It is one specific thing that is being taught... like a code of conduct or the late fees of the library. So you can't really belehren someone about biology... I wouldn't know how to understand the following sentence:
This is so vague that it could be anything... but it certainly is not the biological knowledge itself. Belehren also has a negative touch to it... I wouldn't appreciate being belehrt. I'd prefer to be informiert.
Lehren can take 2 direct accusative objects... the thing being taught and the person being taught.
Lehren doesn't imply completion. It can but it doesn't have to. It can be a life long process. It is also the word for the profession... to teach. Also lehren is not negative sounding. It is pretty neutral. In daily life, I think lehren as a verb is on the decline. I find it kind of weird in a sentence and I would opt for unterrichten. However, the nouns derived from lehren are totally part of everyday talk. Lehrer, Lehrling, Lehrplan, Lehrbuch, Lehrfilm, Lehrvideo, Lehrgang...