Is it permissible for a teacher in a village to take olive oil from the residents if he does not know whether it is zakat or a gift, and is it permissible for him to take it if it is zakat?

Firstly: Gifts from villagers to a teacher:

If the gift is from someone who has students studying with the teacher: It is generally impermissible to accept it to avoid favoritism and injustice among students. If the gift is from someone who does not have students, or from the general villagers as a gesture of honor: There is no harm in accepting it, provided it does not affect the teacher's work or relationship with the students.

Secondly: If the matter is between zakat and a gift:

If you are eligible for zakat and the gift is lawful: There is no need to ask, and you may accept it. If you are not eligible for either: It is not permissible for you to take it, and you should politely decline. If you are eligible for one but not the other: You should ask to determine what is permissible for you.

Shaykh Ibn Uthaymeen was asked about taking money as a gift that is in fact zakat, and he replied: "If you are eligible for zakat, then it is lawful for you. If you are not eligible for it, then it is not permissible for you to take it from him."

Shaykh Ibn Uthaymeen also warned against the error of someone who pays zakat disguised as a gift or to someone who would not accept it without informing them, explaining that zakat is not valid unless the eligible recipient knows about it and accepts it.