How much fuel does an airliner consume during passenger boarding?
This depends on the aircraft and airport and how they are equipped. The time varies as it takes between 30 and 120 minutes to turn around an aircraft. So ill provide GPH numbers that can be used as desired.

Some airports have umbilicals that can provide ground power. This is often fed via a cable that feeds out of the terminal. This is usually fed from what ever power supply is feeding the airport.  

Some airplanes (many these days) have an APU (auxiliary power unit). These are capable of feeding the power demands during a turn around. The also make it far safer for the ground crew by not having the engines running when there is crew on the ground working around them. The fuel burn of an APU can be found in the answer to this question. 

Some airports supply power through the use of a ground power unit. These can be diesel powered or run on Jet-A and their output varies. They even have them for the small GA aircraft I fly to charge the battery if it dies on the ramp. This article lists a burn of 5 gallons an hour for a GPU although sizes vary. This maker lists their GPU's by Gallon per kilowatt hour which may be more appropriate for matching load demands.