DUNWOODY, Ga.—The special election Tuesday to fill a Georgia House seat has drawn staggering sums of money, a barrage of advertisements and the intervention of the president as Democrats attempt to win a seat that Republicans have held since 1979.

Energized Democrats say victory in Georgia’s Sixth Congressional District would put an early dent in the Republicans’ House majority—and send a signal to the nation about President Donald Trump’s effect on his party’s voters. Republicans, split among 11 candidates, aim to take the...