No, since 2010 the regulations have stated that F1 cars must start with sufficient fuel to complete the race and provide a sample for testing after the race.
Refuelling was banned to reduce costs, by doing away with costly equipment, and to improve safety during the pitstop.
The current hybrid cars start a race with around 110kgs of fuel on board, this affects the car's performance and requires adjustment in both car setup (engine map) and driving technique during the race. You may hear the drivers engineer instructing him to conserve fuel.
Fixed fuel loads are also the reason that non-racing laps (behind the safety car) are counted, as there is only enough fuel to circulate the track for the specified number of laps.