Is the lie told by the questioner's friend—with the aim of enabling the questioner to travel and work—the cause of the bad luck the friend is experiencing, and what should be done in this situation?

The impermissibility of cheating, lying, and deception is well-known in Islam, and sins and transgressions are a cause of calamities. It is obligatory to repent to Allah Almighty and to strive in acts of obedience, because good deeds erase bad deeds. As long as the lie has occurred, and the work is being performed efficiently, there is no impediment to continuing in it, and there is no obligation to inform the employer to prevent harm.