Yes, gas water heaters often drip after they are put into service after a prolonged shut-down, but it is not because of the pilot light.
When the main burner ignites below a tank of cold water, the moisture contained in the gas flame will condense on the outside of the tank bottom and the flue that passes through the center of the tank.
Once the water is hot, the water heater should stop dripping and the moisture below the water heater should evaporate.
Continued dripping may indicate a leak and the water heater may need to be repaired or more likely replaced if it is older--more than five to seven years old.
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