I am going to rent a room from a government school for teaching my students. I will pay rent each month to this school's manager. If the manager doesn't inform the government about this rent and uses it for himself, will my earnings from teaching in this room be halal? Is this considered a bribe?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger. 

The lease contract is not valid unless it was concluded by the owner of the leased property or his representative.

Since the school principal is neither the owner of the school nor a representative [or an agent acting on behalf] of its owner for renting it, then it is not lawful for him to lease the room. If he does so, he is a transgressor and the government thereby has the right to fine him or to fine you as a tenant. If the government imposes a fine on you as a tenant, you have no right to ask the principal to pay you back for the fine which the government imposed on you. If you had paid the principal a fee, you have the right to take it back from him.

Ibn Qudaamah said:

“If a usurper rents out a usurped property [or something that is taken unlawfully], then the lease is void, according to one of the narrations (form Imaam Ahmad), just as selling, and its owner has the right to impose a fine on whoever of them (either the usurper of the tenant) he wishes; a fine equal to the rent of a similar property; so if the tenant was fined, he has no right to ask back for what he had paid, because he entered a contract while guaranteeing the benefit [of the rented property], unless the rent of a similar property is in surplus of what was named in the contract; in which case, he can ask back for the surplus and he becomes exempted from what is named in the contract. In case he had already paid it to the usurper, he is entitled to ask for it back...” 

Some scholars believe that the rent money of something taken unlawfully is for the usurper (the school principal) but that he should give it in charity or give it back to whom the property was taken from, which is the government, according to your question.

Badaa’i as-Sanaa’i’ (a Hanafi book) reads, “If the usurper rents what he has taken unlawfully, then he should give the money of the rent in charity because it is money that he had obtained in an unlawful manner, which is disposing of someone else’s property without his permission, so it should be given in charity…”

Bahr ar-Raa’iq (a Hanafi book) reads: “If a usurper rents out the usurped property and then he gives back the property, the fee is for him and he should give it in charity or he gives it back to the one from whom he had taken it unlawfully.”

Therefore, it is not permissible for you to rent the room from the principal as long as you know that he is not entitled to rent it. But if you rented it and used it to teach, then the money that you earned from teaching is lawfully yours as it is not money that has grown from what is taken unlawfully; it is a fee for work that you have done; they are two different things.

Allaah knows best.