The role of Vice President has become much more active since the times of Teddy Roosevelt. As it now serves as the next in line should something happen to the President, the pressure to be prepared for the role of their current boss. They also serve in very similar functions to the President in terms of appearances, diplomatic trips and other social functions that the President simply can not handle on his own in this day an age.
The vice presidency of the United States remains an important position. Despite some who have withered from a lack of involvement in the administration, it is still a heartbeat away from the presidency, and thus very important. More recent occupants, such as Joe Biden and Dick Cheney have been utilized more.
The Vice President is second-in-command and needs to be ready to assume the office of President at a moments notice. The Vice President role should be expanded and allowed to be more powerful and effective. There have been many times in history when the Vice President has taken over so we need to be prepared for that unexpected event.
No, I do not believe that the Vice President position should be eliminated. Despite Teddy Roosevelt's boredom, the Vice Presidency is an important position in the administration. If a Vice President is "bored" I believe they should find opportunities to serve the public good instead of bettering themselves on a personal level, like Roosevelt's consideration to finish law school.