0707,
Oil refining is one of the least labor intensive industries, so one would think outsourcing is not a major issue. Still I see that more and more support processes are outsourced, especially those that can be pretty well defined on an administrative level and require little face-to-face contact, like purchasing, accounting or IT. (I don't necessarily agree with that and will be the first to confirm that a distant accounting or IT dept is a major pain in the butt!)
Re the refining process itself, the European diesel production is already being "outsourced" to a certain degree because of the growing unbalance between diesel and gasoline production in Europe. Excess gasoline or gasoline components are shipped to the US, the shipping cost is not prohibitive. I would have difficulty to believe though that Europe would accept closure of a significant number of European refineries and overdependance on supply from other countries, refining is just too strategic and you don't move refineries around like call centers.
Re impact on the final customers (Joe sixpack who fills up his truck), they won't care, with all the swapping and trading they already don't know what they've got in their tanks in the base case.