If someone who is renting or hiring some facility or service stops making use of it with no excuse, is he obliged to pay the fee in full?

If the contract between you states that payment is to be made on a monthly basis, then she must pay for the month in full, because the contract of hire is binding on both parties, and the one who does not make use of the service for which he signed the contract is the one who is at fault for no longer making use of the service, so the fee cannot be waived.

It says in al-Mawsu‘ah al-Fiqhiyyah al-Kuwaitiyyah (1/253):

The basic principle regarding contracts of hire, according to the majority of scholars, is that such contracts are binding, and neither party can unilaterally cancel the contract except for a reason on the basis of which a binding contract may be nullified, such as a defect or problem with the item or service in question, or if it can no longer be made use of.

They quoted as evidence for that the verse in which Allah, may He be exalted, says (interpretation of the meaning): {O you who have believed, fulfill [all] contracts} [al-Ma’idah 5:1]. End quote.

Ibn Qudamah said: A contract of hire or rental is a binding contract, which means that the one who is hiring or renting will pay the fee and will make use of the item or service in question. If he cancels the deal before the end of the contract period, and no longer makes use of the item or service in question by his own choice, the contract is not cancelled and the fee is still owed.…

Al-Athram said: I said to Abu ‘Abdillah (Imam Ahmad): A man hired a camel, then when he came to Madinah, he said to him: Cancel the deal for me. He said: He has no right to do that, and he must pay the fee."(Al-Mughni  8/23).

It says in al-Ma‘ayir ash-Shar‘iyyah in the section on hiring and renting: It is permissible to cancel a contract of hire or rental with the agreement of both parties, but neither of them has the right to cancel it [unilaterally] except in the case of unforeseen circumstances.

Moreover, the one who is renting or hiring has the right to cancel because of a fault or defect that appears in the item being rented or hired, which prevents him from making use of it."(Al-Ma‘ayir ash-Shar‘iyyah, p. 141).

If no unforeseen reason arises that prevents making use of the item, and there is no excuse or reason to cancel the contract, then the contract of hire or rental is binding, so long as the driver – in the case mentioned in the question – made himself available to transport the teacher, and committed himself to that in the customary manner, and there was nothing to prevent her from travelling with him.

That is reinforced by the fact that the custom in your country is that once the month has begun, if the user of the service did not withdraw from the contract, then the fee must be paid.

Based on that, the teacher must pay the month’s fee in full, unless you reach an agreement that she will pay for only one week, or less or more than that, and reaching a deal is better.

And Allah knows best.