Based on the definition of Biblical literalism from the Wikipedia page:
Biblical literalists believe that, unless a passage is clearly intended as allegory, poetry, or some other genre, the Bible should be interpreted as literal statements by the author.
I'd have to say no, there is no scriptural warrant for this practice. There is nothing wrong with understanding Bible stories literally; the problem is when we stop there. Passages not clearly labeled as allegory or typology may still be able to teach us something more if we seek a deeper meaning.
Jesus said he taught in parables because:
The reason I speak to them in parables is that "seeing they do not perceive, and hearing they do not listen, nor do they understand.' With them indeed is fulfilled the prophecy of Isaiah that says: "You will indeed listen, but never understand, and you will indeed look, but never perceive. For this people's heart has grown dull, and their ears are hard of hearing, and they have shut their eyes; so that they might not look with their eyes, and listen with their ears, and understand with their heart and turn— and I would heal them.' But blessed are your eyes, for they see, and your ears, for they hear.
His parables had a deeper meaning that could be found only by those who searched for it.
And as your quote from John 5:39 states, Jesus expected his followers also to search the scriptures to see how they testify about him. (At that time, "scriptures" referred to what Christians today call the Old Testament.) In many cases this involves going beyond the literal, historical meaning of the words. Many New Testament writers uncovered hidden meanings in Old Testament passages:
Matthew, for example, found these:
"Look, the virgin shall conceive and bear a son, and they shall name him Emmanuel," which means, "God is with us." — Matthew 1:23; compare Isaiah 7:14
"Out of Egypt I have called my son." — Matthew 2:15; compare Hosea 11:1
The first, in its original context, referred to a child that would be born in Isaiah's own time, as a sign to King Ahaz that Israel would prevail over the two nations that were threatening them.
The second, in its original context, referred to the Exodus and to God's love for his people and his care for them despite their unfaithfulness. But Matthew found a Christological meaning in both of these phrases.
Paul, in Galatians 4, saw the birth of Isaac as a foreshadowing of Christianity:
For it is written that Abraham had two sons, one by a slave woman and the other by a free woman. One, the child of the slave, was born according to the flesh; the other, the child of the free woman, was born through the promise. Now this is an allegory: these women are two covenants. One woman, in fact, is Hagar, from Mount Sinai, bearing children for slavery. Now Hagar is Mount Sinai in Arabia and corresponds to the present Jerusalem, for she is in slavery with her children. But the other woman corresponds to the Jerusalem above; she is free, and she is our mother. For it is written, "Rejoice, you childless one, you who bear no children, burst into song and shout, you who endure no birth pangs; for the children of the desolate woman are more numerous than the children of the one who is married." Now you, my friends, are children of the promise, like Isaac.
Hebrews 7-10 is a long passage that shows how Melchizedek foreshadows Christ, and how all the ceremonial acts of Old Testament worship, including God's covenant with Israel, the tent of meeting, and the law itself, are shadows and symbols of what was to come through Christ's sacrifice.
And finally, Jesus himself applied a Christological meaning to Jonah in the whale:
But he answered them, "An evil and adulterous generation asks for a sign, but no sign will be given to it except the sign of the prophet Jonah. For just as Jonah was three days and three nights in the belly of the sea monster, so for three days and three nights the Son of Man will be in the heart of the earth. — Matthew 12:39-40
So, rather than limiting themselves to the flat, historical meanings of the Old Testament passages, Jesus and his disciples (and the New Testament writers) consistently searched the scriptures to find their testimony about Christ.