I use arnica cream for everything and the results are very good. Bruises, cuts, scrapes and bites even sunburn heal quite quickly. It contains Aloe Vera too. I don't know if it will help with the oil burns but it's worth a try seeing as it works well healing skin.
Lavender oil is also very good, I use this alot too. And honey has natural healing ingredients and is also very soothing. Another one is full fat butter; I always dap butter on a burn and scald and I do get relief. If the burns are really that bad though perhaps you should see a doctor or check out a professional forum.
Whatever you do, don't scratch and maybe take a pain killer to help ease the discomfort. Oh! I just had another idea; what about an Aloe Vera face mask or make one from turmeric, I know turmeric is full of healing ingredients. Make a paste with the spice and a little water and gently place over the burns and leave for as long as you can.
I sincerely hope you feel better soon and that you don't have scars.