Tell me please which sentence is correct and why.
If any of them isn't correct, the teacher will help you.
If either of them isn't correct, the teacher will help you.
And by the way, is it either of them is or either of them are?
Tell me please which sentence is correct and why.
If any of them isn't correct, the teacher will help you.
If either of them isn't correct, the teacher will help you.
And by the way, is it either of them is or either of them are?
Either of your suggested sentences could be correct, depending on the context. If there are exactly two problems or things to be checked, that is if "them" consists of exactly two items, then "either" would be correct. If "them" consists of more than two items, "any" would be correct.
While "any" can be used for a group of two, it usually isn't so used.
Strictly speaking, "either" means "one or the other", that is, it indicates a choice of exactly one of two things. "You may either go or stay." and so the use of either suggests that there is no chance of both things being not correct. One could say
If either or both of them isn't correct, the teacher will help you.
But this is a bit on the formal side, and where it is clear that "both" is possible, it could be omitted and implied.
And by the way, is it either of them is or either of them are?
It should be "either of them is". Because either indicates a choice of one option, it takes singular forms. It should also be "any of them is.
As the this post about the use of "each" says, singular forms should be used here, although most people will not strongly object to the incorrect plural forms. While "them" represents a group, "either", "any" (and "each" not used in the question) consider the members of that group one at a time, so singular forms are correct, and plural forms are not.
"Any of them" can be thought of as short for "any one of them" or "any one or more of them". Therefore one should say:
If any of the answers is incorrect, the teacher will help you.
One should not say: "If any of the answers are incorrect..."
Another acceptable alternative would be:
If any answer is incorrect, the teacher will help you.
I think it would depend on what the pronoun "them" represents. In either case, you would want to use aren't, not isn't. The correct sentence to use depends on whether "them" refers to two objects or more than two objects.
"Them" would be a plural object pronoun. Suppose you knew that "them" referred to "the questions" (more than 2 questions). In this case, you would need to have your noun and conjugated verb ("to be" is the infinitive) show quantity agreement.
If any of the questions aren't correct, the teacher will help you.
If "them" refers to exactly 2 questions, you would be trying to say
If this question isn't correct or this question isn't correct, the teacher will help you.
This implies a two-choice scenario, and you would say
If either of them aren't correct, the teacher will help you.
Another possibility is when you don't know what quantity "them" represents. Suppose you don't know how many "questions" are being described. It could be 2, 10, 100... If the quantity is unknown, you would say
If any of them aren't correct, the teacher will help you.
On top of all of this, you could also substitute "aren't correct" with "are incorrect".
If any of them are incorrect, the teacher will help you.
If either of them are incorrect, the teacher will help you.