Ummm, this one is complicated.

Mostly CO2 dosen't disolve into water (hence all the bubbles still in the cola can). Little of it does forming a very short lived carbonic acid species. It can be decribed by the reaction below.

CO2 +H20 -->H2CO3

This makes

H2CO3 --> H+ + HCO3-

and

HCO3- +H20--> H+ +CO3-

But when one would drink cola with bicarbonate, and carbonate ions present, they are the ones that will react with the oils and thus "deprotonate" them and make them soluble. Similarly with oil spots on the driveway or sulfuric acid on the battery terminals of a car battery.

However, cola seems to be quite an interactive system. There are sugars, phosphates, carbonates, and organics present, that create some sort of chemical degreasing action. Plus there is the added bonus of the effervescent quality that physically breaks up the alaphatic hydrocarbons and their hydrophobic clumpyness.