Well technically no. You could reject much of what's recorded in the Bible and still have faith and be saved. It's not faith in the Bible that's required, it's faith in Jesus.
What you can't reject is that you're a sinner, and that God sent his only begotten son, Jesus, to pay the price for your sins. And you need to accept Jesus as your Savior through prayer.
Rejecting the rest would probably result in many heretical beliefs, but you could still be saved.
Generally, faith in the Bible and faith in Jesus go hand in hand. But it's entirely possible for that to be largely unhinged.
God has revealed Himself in three ways, according to Romans 1 and Hebrews 1:2:
- through what has been created
- through His Word, the Bible (all of it) &
- in His Son
As far as the Bible, Jesus said in Matthew 5:17-19:
“Do not think that I came to abolish the Law or the Prophets; I did not come to abolish but to fulfill. For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass from the Law until all is accomplished. Whoever then annuls one of the least of these commandments, and teaches others to do the same shall be called least in the kingdom of heaven; but whoever keeps and teaches them, he shall be called great in the kingdom of heaven.”
Then there is the issue of false prophets that He addresses by a comparison of a bad tree (the false prophets) and a good tree. He said you will know them by their fruits.
(Matthew 7:15-20)
Then there is His teaching on those who call Him “Lord,” but won’t enter the kingdom of heaven because they don’t do His well; they don’t act on His words, but instead practice lawlessness. (Matthew 7:2-29)
Jesus is the living Word.