IronWill summed it up. Here are some Scripture references to those issues for which a specific passage of Scripture comes to my mind.
Purgatory
The Bible says that the judgment immediately succeeds death, without any need for purgatory.
And just as it is appointed for man to die once, and after that comes judgment, (Hebrews 9:27)
This Scripture also seems to suggest the same:
For we know that if the tent, which is our earthly home, is destroyed, we have a building from God, a house not made with hands, eternal in the heavens. For in this tent we groan, longing to put on our heavenly dwelling, if indeed by putting it on we may not be found naked. For while we are still in this tent, we groan, being burdened--not that we would be unclothed, but that we would be further clothed, so that what is mortal may be swallowed up by life. (2 Corinthians 5:1-4)
Elsewhere (1 Corinthians 3:12), the Bible specifically says that
our works will be tried by fire. Catholics have misinterpreted the Scripture to say that we also will be tried.
Papal system
The Bible says that Jesus himself is the head of the church:
And he is the head of the body, the church. (Colossians 1:18)
This one is a bit tricker, since the Bible does say that Peter was the head of the early church. I know some believe that Peter's confession was the rock on which the church, but as R.C. Sproul's commentary observes, but for Roman Catholic papal abuses, Christians would probably have never abandoned the passage's obvious reference to Peter. The problem for Catholics is that the Bible
does not teach Apostolic succession. There's nothing in the Bible to suggest that Peter's authority didn't die with him.
Requirement for priests to be celibate
The Bible teaches us not to forbid marriage.
Now the Spirit expressly says that in later times some will depart from the faith by devoting themselves to deceitful spirits and teachings of demons, through the insincerity of liars whose consciences are seared, who forbid marriage and require abstinence from foods that God created to be received with thanksgiving by those who believe and know the truth. (1 Timothy 4:1-3)
In all fairness, the celibacy requirement is not a dogma of the Catholic Church. It's more of a Western tradition. To this day, Eastern Catholic priests, who are in full communion with Rome, are permitted to have wives (provided that they marry before their ordination). And when some married Anglican priests joined the Catholic Church due to the homosexuality controversy, they were fully accepted. Still, it's not a good idea for the church to encourage celibacy among people who aren't called to it, especially when the Bible says that people in ministry are not necessarily called to celibacy (1 Corinthians 9:5).
Multiple Mediators between God and Man
The Bible says that Jesus is the only mediator between God and man:
For there is one God, and there is one mediator between God and men, the man Christ Jesus, (1 Timothy 2:5)
Confession of sins to priest
This one isn't unbiblical, per se. The problem is not the encouragement to confess sins to a priest, but the requirement that you can confess sins
only to a priest. The Bible does command the confession of sins to other Christians:
Therefore, confess your sins to one another and pray for one another, that you may be healed. The prayer of a righteous person has great power as it is working. (James 5:16)
Biblically speaking, confessing sins to any other Christian should be equally valid as confession to a priest. This tradition arose because in the early church, Christians would sometimes confess sins to people who would later blackmail them with the information (I suppose there were a lot of false brothers around back then, just as there are today). So pastors began receiving
all confessions. The problem for the Catholic Church arose when they replaced the commandment of God with their human tradition.
Anyway, those are some reasons I can think of as to why Catholic doctrine is at odds with the Bible. I'm not as hardcore anti-Catholic as other Bible believing Christians, in that I don't think the Catholic Church is quite apostate (though it appears to be getting there, albeit slowly). But given Catholicism's apparent disdain for evangelism, I do think that the Catholic Church is defunct.
