Your English teacher is right that "Take it with yourself." isn't idiomatic.
But I don't think it is because of the verbs "take" and "bring", because we can say:
"Take one for yourself too if you want."
"We nearly brought ourselves to tears with our sad stories."
"I can bring it myself."
It may be that it is the preposition "with" that causes the reflexive (-self) pronoun to sound "wrong". Here are a couple of examples using other verbs and "with":
"I still keep his photograph with me." (not "myself")
"Carry this with you at all times." (not "with
But on the other hand,
"I don't have my passport on me."
"I still see that terrible image before me."
Neither of those sounds right with "myself", and they don't contain "bring"/"take" or "with".
My conclusion is that it is the concepts being discussed, and not any of the individual words, that cause the reflexive pronoun to be appropriate or not.
The only examples I could think of (where we can't use a reflexive pronoun) do involve the idea of accompaniment or possessing/having something near. Perhaps in English that is a concept that doesn't permit a reflexive pronoun. (?)
Yours is an interesting question.