Final answer:
The part of a high efficiency furnace that should be tested if the main burner shuts off after ignition is the flame sensor, as it is likely causing the safety shutdown.
Step-by-step explanation:
If the main burner of a high efficiency furnace shuts off after initial ignition, the component that should be tested is the flame sensor. The flame sensor is responsible for detecting whether the burner is lit and ensuring that gas keeps flowing as long as there is a flame. If the flame sensor is dirty or faulty, it may incorrectly report that there is no flame, thus causing the gas valve to shut off as a safety measure. It's not likely the thermostat, as it triggers the initial ignition. The igniter has already done its job by igniting the burner, and the gas valve was open to supply the gas. So, the most logical part to check is the flame sensor.