Henry's idea is good. Another option to get up most of the oil, is to pour kitty litter on the area and walk on it for a few minutes. Grind the cat litter into the oil marks on the floor and wait a few minutes. When you sweep up the litter, the floor will no longer be oily.
This is a simply way to pick up the oil in minutes, and it is less hazardous than using TSP. However, if you wanted to paint the floor, I would recommend TSP or even muric acid. If the old oil dripper is sold and you want to throughly clean one time, try the TSP. However, I would still pick up most of the oil with cat litter first.