We usually think about (our) wrath-deserving sin as transgression, as leaping over the fence He forbids us from jumping. Our spiritual death certainly activates transgressions (see Ephesians 2:1), but our spiritual death also animates unbelief. Unbelief is no less a wrath-deserving sin.
Unbelief deserves wrath because, whether we would say it like this or not, unbelief questions God’s honesty, His trustworthiness. God never lies (Titus 1:2) and He never fails to fulfill His promises (Romans 11:29). I don’t appreciate when my kids don’t believe me, but how much more wrong is it to doubt God?
Unbelief also warrants judgment in light of the many evidences God has given. He graciously substantiates His claims, even to us doubting Thomases. “Put your finger here, and see my hands; and put out your hand, and place it in my side. Do not disbelieve, but believe” (John 20:27). He provides many proofs of His believability and men will be judged according to the revelation they’ve disbelieved.
The sin of unbelief also earns wrath as direct disobedience to the commands to believe. Faith is important because it is necessary for salvation. But faith is also imperative, so it is necessary for obedience. Disbelief is disobedience.
We who call ourselves believers, who profess faith, cannot let our belief slump or slip or wane. God still is trustworthy, He still manifests sign upon sign, and He still commands us to believe.