We had this problem and were advised to paint over the oily mark with clear nail varnish and then paint over the top otherwise it just kind of seeps back through. Blu tac on painted walls is an evil thing to get rid of!
On 'this morning' they rolled up a white piece of bread into a ball and gently rubbed/dabbed it on the greasy mark. I haven't tried it so dont know how well it works but that was there tip on how to get it off
It did. The person who gave us the tip had a lot of experience with moving people in and out of army accommodation and came across the problem a lot. The other big tip was Steradent tablets dissolved in water for removing stains from the carpet. Oh and a can of cola down the loo to clean it!