You know, I've heard this much of my life, but I cannot find any support for this in Scripture. Rather, believers are admonished to treat unbelievers as if they are already dead, and go their way without looking back, even if the unbeliever is a close relative (father.)
The way God looks at it, the default position of every man is justifiable judgement. Why should it be any different for us?
I'm not saying that we shouldn't expend our lives reaching the lost, but that should not be the center of our focus.
The reason why it should be different until the dying moment is because undeserved grace was pressed upon us, therefore we are always right to be part of the process of pressing it upon others. With care of course, that we don’t get so torn up with refusers of pearls that we can’t minister anywhere else. Jesus had infallible perception of who would never believe, and we do not, and therefore we would be exercising hard hearts by taking that attitude towards all unbelievers.
Also “God doth take no pleasure in the death of the wicked.” The sadness does not need to drag us down forever, however to express it in some measure is not an unscriptural thing (as long as we don’t let negative scriptures completely blot out our perception of positive ones).