In the story of Thomas and the risen Christ (John 20), Thomas, one of Jesus 12 elect, doubts when the other 11 disciples tell him that they have seen the Lord. He says that he will not believe them unless he actually sees his wounds and touches them.
This tells us that Thomas did not just believe anything that he heard, and also that he needed to see (observe) and touch (examine) Christ up close himself before he would believe.
Now, did God scold him for this, or cast him away, or did he understand that Thomas needed to be sure in order to believe? The latter is the truth. Christ reveals himself as risen to Thomas, and tells him to see (observe) his wounds and touch (examine) them as well. And he does this, in order that Thomas should no longer doubt, but believe.
Believing does not necessarily mean that we just believe. Even though Christ suggests that it is blessed to believe without having seen, he does not say that it is blessed to believe without having first examined the Scriptures. That would actually be rather foolish.
Today, people might not get to see and touch Jesus in that same way that Thomas did, but since he is the living Word of God, the Holy Spirit also lets us closely observe the crucified and risen Christ today.
So in my view, when it comes to the Bible, it is not necessary to examine Scripture because I doubt that it's true, but it is necessary to examine Scripture if I'm uncertain about something that I've just heard about or thought up myself.