Article V ("On the Sufficiency of the Holy Scriptures for Salvation") of the Articles of Religion of the United Methodist Church says this:
The Holy Scripture containeth all things necessary to salvation; so that
whatsoever is not read therein, nor may be proved thereby, is not to be
required of any man that it should be believed as an article of faith,
or be thought requisite or necessary to salvation. In the name of the
Holy Scripture we do understand those canonical books of the Old and New
Testament of whose authority was never any doubt in the Church.
I affirm this while also recognizing the Wesleyan Quadrilateral, which aligns with prima scriptura, not sola scriptura:
"Scripture is considered the primary source and standard for Christian
doctrine. Tradition is experience and the witness of development and
growth of the faith through the past centuries and in many nations and
cultures. Experience is the individual's understanding and appropriating
of the faith in the light of his or her own life. Through reason the
individual Christian brings to bear on the Christian faith discerning
and cogent thought. These four elements taken together bring the
individual Christian to a mature and fulfilling understanding of the
Christian faith and the required response of worship and service." (from the UMC website, emphasis mine)
However, while both of these are related to the role scripture should have in our lives, neither one directly touches on the idea of infallibility. For scripture to be "the primary source and standard for Christian doctrine," it must be God-inspired; the Spirit must be able to work in us through scripture.
Moving from there to the word "infallible" sometimes makes me uncomfortable, though, because it has been used with a variety of meanings - sometimes equated with "inerrancy," sometimes stronger than it, sometimes weaker, sometimes used to say that the Bible should be taken literally. Wesley describes Scripture as infallible in his 16th sermon:
"The same truth (namely, that this is the great means God has ordained
for conveying his manifold grace to man) is delivered, in the fullest
manner that can be conceived, in the words which immediately follow: All Scripture is given by inspiration of God; consequently, all Scripture is infallibly true; and is profitable for doctrine, for reproof, for correction, for instruction in righteousness; to the end that the man of God may be perfect, thoroughly furnished unto all good works (2 Tim. 3:16, 17)."
First of all, it's worth noting that Paul was referring to Hebrew scripture (as that was the scripture that existed), which adds some interesting context. Paul says in Romans (repeatedly) that we are no longer bound by the law but also that the law taught us what is sin. The law is holy, the law is true, and it is "profitable for doctrine, for reproof, for correction, for instruction in righteousness," but we are not slaves to it. The principles in the law are holy principles from which we can learn, even though we are under a new covenant of grace.
When someone uses Biblical infallibility as a synonym for Biblical inerrancy or to mean that Biblical literalism is the "right" interpretation, I'm not on board. The example that's been on my mind recently, thanks to Amy-Jill Levine's Old Testament lectures, is that of King David. There's not very much archaeological evidence for David, and even if King David existed, he probably didn't do all the things that we ascribe to him. That doesn't mean I don't think these stories are useful; in a sense, I even think these stories are true. I believe they're true in a way that has nothing to do with whether or not the events occurred as recorded. They're true in the way that Paul describes in the letter to Timothy.
Here's another (perhaps more typical) example related to literalism and inerrancy. Creation as told in Genesis 1 establishes God as Creator of everything. It's an origin story of the world and everything in it, of order, and of the Sabbath, with God present and working at the very beginning. I tend to identify as a theistic evolutionist, so I don't take the creation story literally, but I still believe it is true.
That phrasing can be more confusing than infallibility, though, so sometimes I make a distinction between 'true' and 'truth' and say that while I don't believe something is true, I think it has truth. I don't find that particularly comfortable, though. All this is sort of sticky, honestly.
The really key thing for me is that we can't ignore context - both the context in which scripture is written and our own contexts. The tradition, experience, and reason parts of the Wesleyan quadrilateral are very much about our own contexts. As for the context in which scripture is written, scripture as a whole tells a more complete story than do its parts in isolation; that's why the writers of the New Testament books quote the Old Testament so much. There is no new covenant without the old covenant. We do not become God's set-apart people without Israel first being God's set-apart people.
As a different and more specific example (also more typical, yet again), Paul on women! Here's 1 Corinthians 14:34-35:
should remain silent in the churches. They are not allowed to speak, but must be in submission, as the law says. If
they want to inquire about something, they should ask their own
husbands at home; for it is disgraceful for a woman to speak in the
Ouch. This from someone who elsewhere praises some women who were teachers in the church. The local context is that this is in a section about orderliness in the church, especially related to speaking in tongues. Paul is trying to restore order to an unorderly church. In churches women were praying and working with men in ways that they didn't normally in society, and Paul doesn't speak against that here. There's a sort of revolution going on, and Paul is pulling it back a bit. This section is about the women, but he's calling overall for greater order and constraint in the church. (Based on a relevant section from Disciple 1 Bible Study)
Do I think this bit is the right response? Do I think "women should remain silent in the churches" and "ask their own husbands at home?" No, not at all. Do sections like that one still make me nervous with the idea of infallibility and God-inspiration? So much. But I understand the context and the principle underlying this - order - and I believe Paul's message about that principle to be God-inspired and true. Other parts of scripture show women faithfully serving in all kinds of roles among God's people.
When people talk about the Bible as infallible, I get uncomfortable because without clarification that can mean so many different things. I don't interpret the Bible literally, and I don't consider it inerrant (even before translation). But at the end of the day, I believe the Bible is God-inspired, and I believe it is truth, and so as far as Wesley's usage goes, I believe in Biblical infallibility.