The oil travels back to the heart and gets pumped to your lungs, where it will cause an accute reaction (coughing fits, nausea, fainting, etc). Once you get through the first couple minutes (and I've never heard of anyone actually dying from this), you'll be fine.
Theoretically, if you injected so much oil that your lungs couldn't clear it all by coughing, the rest would return to the heart and be pumped to the brain (among other things), where it could cause an ischemic (lack of oxygen) stroke. But I doubt you could ever inject enough oil to accomplish that, since you'd start coughing and freaking out way before you could inject it all.