If I recall correctly from my Immunology Class, Naive B Cells become 'activated' when it recognizes a pathogen (it's antigen), and it undergoes proliferation and differentiation in the lymph nodes to form Memory B Cells and Plasma B Cells. The Plasma B Cells exit to various parts of the body to deal with the pathogen by secreting antibodies and these antibodies deal with the pathogen by different means: opsonization/phagocytosis, complement, etc.
naive B cells generally last 3 weeks. If they aren't activated they die.
plasma cells (activated naive B cells), last an approx 3-4 weeks (not 100% sure of range) and continue to secrete antibodies during that period.
But the antibodies they secrete, depending on which the type, can last anywhere between a few weeks to a few months. The antibodies themselves though (which are proteins) definitely don't last forever (like all proteins). Eventually certain suppressor cells kill off plasma cells after the response has taken its course, and all that remains are a few memory cells, which are reactivated again should the pathogen invade the body in the future. These memory cells are the reason why the second and later response are always much faster because the initial response.
Also, unlike naive B cells which don't last more than a few weeks, naive T cells last a long freakin time and the reason for this is because of the involution of our thymus through adulthood. T cells are essential to activating B cells and without them, we'd definitely be in an immune suppressed state.