They most certainly do. I just booked BKK-xSIN-oMNL and return on SQ in F using BD miles. UA wouldn't book it becuase it was over the award maximum permitted mileage (although UA did show the availability).

I did it last year using UA miles and all I paid were the miles and the departure tax out of Bangkok (around US$20.00 at the time).

Using BD miles, it was around £82.00 (US$165.00) and, of course, the £3.00 "convenience" fee for booking with a credit card instead of a debit card.

BD doesn't call it a fuel surcharge, but that is exactly what it is.