I imagine Massachusetts state law doesn't require it. LBJ ran for VP in '60, and kept his Senate job just in case. There have been several others.
Lieberman ran concurrently for the Senate and vice president in 2000. Frist should tell these two (Kerry and Edwards) to resign their seats but he won't. In Mass, they are trying to change the law so that the governor, Republican Mitt Romney, would not be allowed to appoint Kerry's replacement should he win.