To stress what halothane said -- it can depend on which carrier and which FF program you're using.

General rule seems to be that for award travel if you're using a US FF program, even though you may be actually flying on a non-US carrier, you're less likely to pay a fuel surcharge. For example, I checked out a Germany to Australia routing on One World (Cathay Pacific via London) on AA and on a European OW FF program. Same routing, same flights, tax/fees on AA was under $150 but on the European FF program was ~$500!

Similarly, most US carriers don't include a fuel surcharge while most European carriers do. If you're purchasing a ticket (rather than redeeming an award), check out alternative carriers, e.g., Delta from US to Paris versus Air France on the same route, and you'll probably find the Delta ticket to be much less than the Air France ticket for the same class/cabin.