Are bloodline curses due to unconfessed sin biblical?

In the charismatic movement of South Africa, we learn that curses need to be broken and often times they are passed down in the bloodlines as a result of unrepented sin. Can unconfessed sin leave a door open for curses to take effect in your life?

The charismatic religious movement is replete with false teaching and demonic activity. The specific teaching you mentioned concerning "curses" is typical of charismatic heresy and is contrary to the Bible's teaching on grace.

The Bible acknowledges that all believers sin continually this side of Heaven (as Romans explained), and therefore we must depend on God's grace in Christ. Because of our faith in Him, we receive the righteousness of God, and therefore, we are never cursed, condemned, judged or otherwise punished for our sin, since Christ took such things for us:

Gal. 3:9 So then those who are of faith are blessed with Abraham, the believer.
Gal. 3:10 For as many as are of the works of the Law are under a curse; for it is written, “CURSED IS EVERYONE WHO DOES NOT ABIDE BY ALL THINGS WRITTEN IN THE BOOK OF THE LAW, TO PERFORM THEM.”
Gal. 3:11 Now that no one is justified by the Law before God is evident; for, “THE RIGHTEOUS MAN SHALL LIVE BY FAITH.”
Gal. 3:12 However, the Law is not of faith; on the contrary, “HE WHO PRACTICES THEM SHALL LIVE BY THEM.”
Gal. 3:13 Christ redeemed us from the curse of the Law, having become a curse for us — for it is written, “CURSED IS EVERYONE WHO HANGS ON A TREE” —
Gal. 3:14 in order that in Christ Jesus the blessing of Abraham might come to the Gentiles, so that we would receive the promise of the Spirit through faith.

Therefore, a believer can never be "cursed" by God, for a curse means eternal damnation, and Christ took that curse for us. Moreover, we have been forgiven of ALL sin already:

1John 1:8 If we say that we have no sin, we are deceiving ourselves and the truth is not in us.
1John 1:9 If we confess our sins, He is faithful and righteous to forgive us our sins and to cleanse us from all unrighteousness.
1John 1:10 If we say that we have not sinned, we make Him a liar and His word is not in us.

John says if we confess our sins (i.e., if we repent and come to faith in Jesus), then He forgives us of our sins. This forgiveness is not conditional on confessing every specific sin we've ever committed, because it is literally impossible for a believer to recognize and confess every sin. We all commit more sins than we even know!

John is saying that by our faith in Christ alone God is just to forgive us of all our sins, past, present and future, by His grace, so that:

Rom. 8:1 Therefore there is now no condemnation for those who are in Christ Jesus.

This is the meaning of grace, and the notion that God would cast "curses" on believers when we sin is the antithesis of grace and love. It is, however, the natural outworking of the charismatic movement's false, works-based theology. Some churches may talk of God's grace, but the dominant doctrines and practices of the movement emphasize works of one kind or another in order to remain in God's love.

Clearly, this is not the Bible's teaching, since no work brought us God's redemption in the first place, and therefore no work "keeps" us there either.

Gal. 3:1 You foolish Galatians, who has bewitched you, before whose eyes Jesus Christ was publicly portrayed as crucified?
Gal. 3:2 This is the only thing I want to find out from you: did you receive the Spirit by the works of the Law, or by hearing with faith?
Gal. 3:3 Are you so foolish? Having begun by the Spirit, are you now being perfected by the flesh?

We highly recommend believers steer clear of such movements and their false doctrines. Instead, seek sound, Bible churches that understand and live out grace.