If your bowels are working well and are clean before you flush and you take the Epsom Salts as in the Hulda Clark flush the oil doesn't stay in your system long.
There is a chance a Liver Flush *may* make skin breakouts worse for a few days afterward but that's due to the release of trapped toxins - not the olive oil. Better out than in. If this 'overload' happens it will quickly pass in few days.
A sluggish or clogged liver can cause all kinds of skin issues. A series of flushes will Definitely improve your skin: less oily, less breakouts, better tone.