to examine something closely to find out whether it is genuine, the way an assayer tests metal
This is the verb of expert scrutiny. It does not mean to put someone through an ordeal to see what happens (that is a different word) (נסה); it names the careful inspection that follows, the moment of looking hard at something and pronouncing whether it is the real thing. Its natural picture is the metalworker who takes silver or gold out of the fire and judges its purity, and the Bible often pairs it with the verb for refining (צרף) to complete that image.
In actual use the word lives almost entirely in the world of God and the human heart. By far its most common setting is God looking inside a person, past the surface and into the hidden seat of thought, motive, and conscience that no other examiner can reach. Worshippers in the Psalms and the prophet Jeremiah invite this inspection precisely because it is so penetrating: they want the one examiner who cannot be fooled to render a verdict on what is really there.
The metal-testing color comes and goes with the context. Where the surrounding words speak of furnaces, silver, and gold, the verb takes on the full force of the assayer weighing purity. Where there is no such cue, it means simply to examine or scrutinize. The same act can also be aimed at speech (the ear that weighs words to tell the true from the false) or, in a few pointed reversals, turned the wrong way around, with people presuming to put God himself to the test.