How do Christians that do not believe in inerrancy interpret the Bible?

Within the Orthodox Church, we believe that the Church is the only proper locus for interpreting Scripture, principally through the writings of the Church Fathers as well as through the texts of the Services, many of which are over 1,000 years old.

Anticipating objections to this, Vincent of Lerins wrote the following in the early 5th century:

But here some one perhaps will ask, Since the canon of Scripture is complete, and sufficient of itself for everything, and more than sufficient, what need is there to join with it the authority of the Church’s interpretation? For this reason,—because, owing to the depth of Holy Scripture, all do not accept it in one and the same sense, but one understands its words in one way, another in another; so that it seems to be capable of as many interpretations as there are interpreters. For Novatian expounds it one way, Sabellius another, Donatus another, Arius, Eunomius, Macedonius, another, Photinus, Apollinaris, Priscillian, another, Iovinian, Pelagius, Celestius, another, lastly, Nestorius another. Therefore, it is very necessary, on account of so great intricacies of such various error, that the rule for the right understanding of the prophets and apostles should be framed in accordance with the standard of Ecclesiastical and Catholic interpretation.

A Commonitory, "A General Rule for distinguishing the Truth of the Catholic Faith from the Falsehood of Heretical Pravity"

(Note: "Catholic" above refers to the entire Church, not just the "Roman Catholic Church", which was a term coined in English-speaking countries)

Interpreting Scripture outside the Church introduces a interesting paradox in that Scripture does not itself declare which books it comprises.

a slippery slope of interpreting the Bible in a way that favors one's own opinion.

The first thing is to know how to avoid this slippery slope. A Christian who does not believe in the absolute inerrancy of the Bible could say that someone who does believe in biblical inerrancy (and therefore interprets everything in the Bible literally) is imposing his or her opinion on scripture, the opinion being that it is inerrant. A further problem is in agreeing on what the Bible actually says. On any issue, one person could find a passage that, read literally, supports his point of view, while another person can sometimes find a different passage that supports a different point of view.

Among those who do not believe in absolute inerrancy, many agree that while the Bible might be considered inerrant on issues of morals and faith, it is not infallible on matters of science or human history.

In the Catholic Church, tradition and papal pronouncements are what effectively define doctrine and limit the scope for misuse of the Bible. In other cases, it is consensus among bishops or other church leaders on what doctrine is consistent with God's will, as revealed in the Bible.

The doctrine of biblical inerrancy is very recent in Christian history

First, let's put biblical inerrancy into perspective.

Two centuries or so ago, and for all of Christian history before that, not a single Christian church, denomination, or preacher held that the Bible is inerrant. The very idea of biblical inerrancy had never even occurred to anyone.

It was only after the Age of Enlightenment (from the 1650s to the 1780s) that any need was seen by Christians for any such doctrine. This need was perceived only when the rapid development of science in the wake of the Age of Enlightenment began to call into question in many people's minds the scientific accuracy of many stories and statements in the Bible, such as the creation of the world in six days and the Flood of Noah.

So for roughly 1,800 years of its history, all of Christianity looked to the Bible as the Word of God without the need for any doctrine of biblical inerrancy.

Biblical inerrancy is not the same as biblical literalism

Further, biblical inerrancy is not the same thing as biblical literalism. Even the well-known 1978 Chicago Statement on Biblical Inerrancy "allows for figurative, poetic and phenomenological language, so long as it was the author's intent to present a passage as literal or symbolic" (quote from linked Wikipedia article; italics in the original).

In other words, biblical inerrancy does not require a wholly literal interpretation of the Bible.

Biblical inerrancy is not necessary for faith in the Bible

Given that all Christian churches functioned reasonably well through many centuries of Christian history without the relatively recent doctrine of biblical inerrancy, the idea that now, two thousand years into Christianity, faith in God and the Bible will suddenly crumble without an adherence to biblical inerrancy seems rather shaky and unfounded.

Adherence to the doctrine of biblical inerrancy is by no means universal among Christian churches. However, I'll leave it to other answerers to say which denominations do and do not adhere to biblical inerrancy.

Biblical inerrancy misses the spiritual point of the Bible

I can, however, speak for my own denomination, the Swedenborgian Church, on the subject.

From a Swedenborgian perspective, the doctrine of biblical inerrancy simply isn't very useful.

In practice, it is used mostly to assert that the Bible is without scientific and historical error. At other times it is used to assert that particular doctrines are correct because they are claimed to be based on the statements of the Bible.

However, from a Swedenborgian perspective, the Bible was never intended to be a textbook of science and history. Why would God be concerned to teach us about science and history? How would that accomplish God's purpose in speaking to humans on earth?

God's purpose is not to give us correct scientific and historical information. Rather, it is to save our souls and bring us to heaven. So what God wants to convey to humans in the Bible is not material world information, which does not affect our salvation, but spiritual understanding, which does affect our salvation.

The Bible is about spiritual change, not about correct doctrine

Further, the point of the Bible is not to instill correct doctrine. Yes, correct doctrine is important. But the purpose of doctrine in Christianity is to lead to faith in the Lord, repentance, reformation, and new spiritual life.

In other words, God's underlying purpose in the Bible is not to get us to think straight, but to get us to live a Christlike life. Faith and doctrine are merely the doorway to making us into "new creations in Christ" (see 2 Corinthians 5:17).

Biblical inerrancy is a useless doctrine for this purpose because it concerns only the head, not the heart. In the Bible, God is seeking to reach our heart, and to "replace our heart of stone with a heart of flesh" (see Ezekiel 11:19; 36:26).

The original writers of the Bible, whether or not they believed or realized they were writing under divine inspiration, all understood that the main point of their writing was to reach people with a message from God and get people to leave behind their evil and sinful lives and live good and loving lives instead. Jesus himself summed up the entirety of Scripture in these words:

"You shall love the Lord your God with all your heart, and with all your soul, and with all your mind." This is the greatest and first commandment. And a second is like it: "You shall love your neighbor as yourself." On these two commandments hang all the law and the prophets. (Matthew 22:37-40)

That is the whole purpose of Scripture according to Jesus himself: to prompt us to love the Lord our God above all else, and to love our neighbor as ourselves.

The doctrine of biblical inerrancy touches only the head.

What God wants to reach is our heart. And all of Scripture is designed by God to touch our hearts and prompt us to turn to God and live from God's love instead of from our own selfish and materialistic desires.

A more spiritual basis for Bible interpretation

Speaking for my own denomination and theology, we avoid any "slippery slope" (to the best of our finite and fallible human ability) by looking to the Bible not as a textbook of science and history, or even as a textbook of theology, but as a message from God whose purpose is to turn us toward God, faith, goodness, love, and spiritual life.

Our mode of Bible interpretation, then, does not require or rely on inerrancy. Instead, it focuses on the spiritual messages contained in and delivered by the text.