From the Boeing FCTM:
"Tire Failure during or after Takeoff
If the crew suspects a tire failure during takeoff, the Air Traffic Service facility serving the departing airport should be advised of the potential for tire pieces remaining on the runway. The crew should consider continuing to the destination unless there is an indication that other damage has occurred (non-normal engine indications, engine vibrations, hydraulic system failures or leaks, etc.).
Continuing to the destination will allow the airplane weight to be reduced normally, and provide the crew an opportunity to plan and coordinate their arrival and landing when the workload is low. Considerations in selecting a landing airport include, but are not limited to: sufficient runway length and acceptable surface conditions to account for the possible loss of braking effectiveness, sufficient runway width to account for possible directional control difficulties, altitude and temperature conditions that could result in high groundspeeds on touchdown and adverse taxi conditions, runway selection options regarding "taxi-in" distance after landing, availability of operator maintenance personnel to meet the airplane after landing to inspect the wheels, tires, and brakes before continued taxi, availability of support facilities should the airplane need repair."
There is no QRH drill to run for this scenario and the Boeing advice is good. Virgin had a similar incident last year and also elected to continue with the flight.
G-VROC