If a technological advance lowers a firm's production costs, why do prices typically fall? Shouldn't the firm maintain the same price and earn economic profit?

While any firm would like to maintain the higher price and experience economic profit, competition prevents this. Any technological advances (with the exception of discoveries covered by patents and trademarks) are likely to be available to all firms. As these technological advances lower costs, firms lower their prices to compete customers away from other sellers. This process occurs until the market price reflects the long-run equilibrium condition of zero economic profit.