I'm assuming you're talking spots and smears from your door handle incident rather than th contents of the chip pan?
If I'm right, then proceed thusly:
Make up two jugs or bowls, one with clean water and the other with a solution of washing up liquid and water - about an eggcupful to a pint, well dissolved.
Get a collection of cleaning cloths and towels to hand.
Using the corner of a cleaniing cloth soak it in the detergent solution then dab on the oil smear. DO NOT RUB. The principle is to enable the oil molecules to meet with the detergent. The carpet may become fairly wet, don't worry. Kind of massage gently at the stain, with your finger if need be. Then blot up the wet with towel(s). If the oil smear hasn't shifted repeat.
After final blotting, apply clean water and blot, once or twice - judge it by eye, the main thing is to extract as much detergent from the fibres.
Hope this works for you - it did for me when the black oil and aluminium grindings from the door spindle fell on a cream carpet during a bout of DIY.