Is it permissible to pay a sum of money to obtain a job with a higher salary if that does not harm anyone, and did Sheikh Muhammad al-Uthaymeen issue a fatwa permitting that?

If you are qualified for the position, and your pursuit of it does not infringe upon the right of another, and the one who offered you the position is the one who decides your appointment, and you find no means of appointment except by paying a sum of money, then there is no objection to paying it to secure your right, and the sin is upon him. However, if the one who offered you the position is merely an intermediary, then there is no objection to giving him compensation for his effort.