Well, let's say that you have a regular customer at a bar.

They get friendly and loose-lipped with the bartender over the span of months. Eventually they let slip that they plan on killing their spouse for a life insurance payout. At first the bartender thinks they're joking, but it becomes evident that there's an actual plan being acted upon and someone's life is very likely in imminent danger.

Does the bartender have a responsibility to go to the police?

Depends on the country. In some places, there is no legal repercussions for not reporting this to the police; in some, it is an actual crime in itself.