Is it permissible or forbidden to give private lessons to university students sent by a friend who works at the university, knowing that the institution prohibits its employees from doing so?

Your friend must adhere to the condition set by the educational institution against dealing with students in private lessons, as Allah Almighty says: "O you who have believed, fulfill [all] contracts" (Quran 5:1); this is to block the means (sadd al-dhari'ah) that might lead to teachers neglecting their duties. As for your work in teaching students, there is no harm in it if the conditions mentioned in Fatwa no. 43657 are met, and it is not done in collusion with teachers to force students into private lessons.