The Mad American Wrote:Or maybe Logan's Run? That sounds kind of like the story line you mention although I don't think there was an island.
It sounds a little bit like "On the Beach" with Anthony Perkins, Ava Gardner and Gregory Peck. In that, it is shortly after WWIII and radiation from the war has spread across the world killing most of the inhabitants except for a handful of people residing on the beach in Australia.
The Government has issued them suicide pills because word has it the deadly radiation is arriving on the shore soon, and the pills are supposed to prevent a horrible death by killing them peacefully.
Not all inhabitants are behind the great pill plan and question if the radiation is even coming. It was quite dramatic, especially the ending. It was a film that was made in 1959.
Not sure if that is it or not, but it was about suicide and was on a beach, it took place in the near future and was in black and white.