For anyone looking for last-minute dinner reservations during the rush of Boston Restaurant Week, a smartphone might help. With an app from OpenTable Inc., it takes only a few taps of the screen to book the first-available spot at the closest steakhouse or sushi bar. That convenience is free for tech-savvy diners, but can be costly to restaurants that list on OpenTable. Some busy eateries can pay thousands of dollars a month to OpenTable for its reservation service and, as important, to use the company’s technology to manage their restaurants. Now, several start-ups are challenging the reign of the 14-year-old San Francisco company that has become the go-to destination for Web reservations. The company provides reservations and management services to more than 800 restaurants in the Boston area, but has become vulnerable to competition because of its fees.