Should the teacher use the threat of punishment in order to get these innocent students to reveal the culprit? In other words, is it proper for a teacher to force students to tell on those who have damaged school property?
(19) A person who talks about others tells secrets; avoid anyone who talks too much.
But do not give away the secrets of another,
(א) שלא (א) לרגל בחבירו שנא' לא תלך רכיל בעמך
Do not reveal secrets against your fellow, as it says in the Torah, "Do not go about spreading slander among your people." (Lev. 19:16) [R. Moshe ben Ya'akov of Coucy]
(1) If a person who is a witness, sworn to testify, sins by refusing to tell what he has seen or heard about the matter, he must bear the consequences of his sin.
(א) חייב אדם להציל את חבירו בין בגופו בין בממונו ובו סעיף א':
ששמע עכו"ם או מוסרים מחשבים עליו רעה ...אלו עובר על לא תעמוד על דם רעך:
(1) someone who heard that informants are plotting against someone or preparing to ensnare him – and he did not reveal this secret to his friend and tell him – he has transgressed, “Do not stand idly by the blood of your neighbor.”
(א) הָעֵד מְצֻוֶּה לְהָעִיד בְּבֵית דִּין בְּכָל עֵדוּת שֶׁיּוֹדֵעַ... שֶׁנֶּאֱמַר (ויקרא ה א) "וְהוּא עֵד אוֹ רָאָה אוֹ יָדָע אִם לוֹא יַגִּיד וְנָשָׂא עֲוֹנוֹ":
(1) A witness is commanded to testify in court with regard to all pertinent testimony that he knows... The source for this commandment is Leviticus 5:1: "And should he witness, see, or know of the matter, if he does not testify, he will bear his sin."