Do Biblical literalists believe we should gouge our eyes out or cut off our hands?

The answer by @GratefulDisciple covers things well. It is important when faced with a many-layered text to consider what each layer says and how the ultimate meaning is a combination of them all.

• The charge to pluck out eye and cut off hand is meant literally to be taken ironically, to shock people and get their attention. That is a rhetorical device.
• There is a structure that connects many verses in the chapter into a unified story. The verse under scrutiny occupies a certain place in the structure. That is the clue that lets us know the meaning is more subtle and it shades the interpretation.
• The structure points to the spiritual issue in view. It is the inability of asceticism to perfect character. Self-righteousness and self effort cannot save. Only God can save. People in all religions are tempted to fall into this prideful trap. Stoics, self-flagellators, and prophets who cut themselves to get Baal's attention (as in the days of Elijah) are examples.

The Gospel of Matthew defines a harvest pattern and then replicates examples of that pattern throughout the book (at least seventeen times). The pattern also defines the structure of Job, Psalm 23, and pops up elsewhere in the Bible. The harvest pattern has seven steps:

1. Preparation. Readying your tools for a new year of farm work, like readying the oxen, repairing the plow, etc. Spiritually, this is consecration, accepting God as your Lord, early religious and moral training.
2. Plowing. There are two parts: the autumn rain, which softens the ground, and plowing, which follows. Spiritually, this is suffering, with the tears of mourning being the rain.
3. Planting. You plant the seed, which is the gospel. You must keep the birds (Satan) from tricking your mind out of belief in Jesus.
4. Pouring. This is irrigation or rainfall. The Holy Spirit encourages and convicts the heart, bringing courage and comfort. You must keep the hardships and persecutions of life from scorching you with the sun's rays, scaring your heart out of believing.
5. Plucking. If you don't pluck the weeds, the garden is overrun. The Father rearranges your priorities and habits of action. You must keep your hands from being distracted by cares, worries, wealth and desires contrary to faith.
6. Producing. If you succeed in all the previous phases, you reap a harvest.
7. Peace. Once the harvest is in, you enjoy a time of peace.

The above pattern is described in detail in the book https://www.amazon.com/Peace-like-Solomon-never-Knew-ebook/dp/B0B9KL8VNG

The pattern is partly defined by Jesus in the parable of the four soils. The pattern is illustrated in Jesus' temptation in the desert. The preparation is his Baptism by John the Baptist and the descent of the Holy Spirit. The suffering is in his case voluntary, fasting for forty days. The three challenges of Satan are concerning the mind (to trick Jesus' mind into choosing bread versus the Word), the heart (trying to inspire fear in the heart by challenging Jesus to jump off the temple), and the hands (trying to distract Jesus into pursuing earthly glory in exchange for idolatry). Then after Jesus wins the contest, angels attend and feed him (producing a harvest) and he wins that round against Satan, earning him a respite, his peace.

Now we can look at Matthew 18 and see how verses 1-14 conform to the harvest pattern. An additional shade of meaning may be gleaned by comparing the harvest phase to the verses.

1 At that time the disciples came to Jesus, saying, “Who is the greatest in the kingdom of heaven?” 2 And calling to him a child, he put him in the midst of them 3 and said, “Truly, I say to you, unless you turn and become like children, you will never enter the kingdom of heaven. 4 Whoever humbles himself like this child is the greatest in the kingdom of heaven.

5 “Whoever receives one such child in my name receives me, 6 but whoever causes one of these little ones who believe in me to sin, it would be better for him to have a great millstone fastened around his neck and to be drowned in the depth of the sea.

Temptations to Sin

7 “Woe to the world for temptations to sin![b] For it is necessary that temptations come, but woe to the one by whom the temptation comes! 8 And if your hand or your foot causes you to sin, cut it off and throw it away. It is better for you to enter life crippled or lame than with two hands or two feet to be thrown into the eternal fire. 9 And if your eye causes you to sin, tear it out and throw it away. It is better for you to enter life with one eye than with two eyes to be thrown into the hell of fire.

The Parable of the Lost Sheep

10 “See that you do not despise one of these little ones. For I tell you that in heaven their angels always see the face of my Father who is in heaven. 12 What do you think? If a man has a hundred sheep, and one of them has gone astray, does he not leave the ninety-nine on the mountains and go in search of the one that went astray? 13 And if he finds it, truly, I say to you, he rejoices over it more than over the ninety-nine that never went astray. 14 So it is not the will of my Father who is in heaven that one of these little ones should perish. (Matthew 18:1-14)

The whole chapter is an extended parable about little children (18:1-14). This pattern is close to the temptation story and the parable of the soils in its analogies.

1. Preparation: (18:1-2) Jesus calls the children to himself. This leads to consecration.

2. Plowing: (18:3-5) We are told to become like a little child. This humbling is painful due to our pride, a form of suffering.

3. Planting: (18:6) One who causes stumbling is a devil, that snatches the seed like in Jesus' parable of the soils.

4. Pouring: (18:7) Woe to the world. To cause woe is to spread fear in the heart.

5. Plucking: We are told to oppose the flesh and cut off offending eyes and hands. (18:8-9)

6. Producing: Seeking the wandering sheep (18:10-12) - this is the harvest of evangelism.

7. Peace: Rejoicing (18:13-14) is the response of a heart at peace.

When Jesus called the little child to come to him, that child had to leave the side of his or her parents or friends. It was a call to separation, to consecration, to exile and wandering until they can find the church and a new place to stand. Yet to reach a place of peace, we must wander again, searching for other lost people join us, when at last we can rejoice together.

The trio of obstacles of stumble-causers (devils), the world, and our eye and hand (the flesh) are recognized as categories in Christian teaching. Our challenge is to not be devils ourselves or try to oppose the devil on our own strength. The woes must be God’s judgment, not our petty wrath or even our own attempts to mortify our flesh by plucking out our eyes or cutting off our hands. And our cultural rules and severe asceticism are as ridiculous a remedy for sin as self-mutilation. When I was a child, it was always my father who got the magnifier and tweezers to remove splinters. My job was to sit there and not squirm too much. So it is when God performs surgery on our souls. Leave the cutting to him but try not to squirm so much.

Summing it all up, the verse in question about plucking our our eyes and cutting off our hands is trying to focus our attention on God as the one who will discipline and direct us away from distractions. He is the one who is best qualified to plucks things out of our lives, not us.

Short answer

Without getting into the weeds about literalism, taking the OP's definition given by evangelicals "as is" (as explained in the GotQuestion article What is biblical literalism), the answer is quite easy: no, it's "obviously" figurative because Jesus (the fulfiller of the law, however that's understood, see Five Views on Law and Gospel) obviously has great respect for the law, and there is a law against self-mutilation in Lev 19:28 (ESV):

You shall not make any cuts on your body for the dead or tattoo yourselves: I am the LORD.

Source: another GotQuestions article What does the Bible say about self-harm / self-mutilation / cutting?.

Generalizing the answer

You ask:

If so, what are the main arguments made by Biblical literalists that it is obviously figurative language?

Evangelical hermeneutics privilege "author-centered" meaning. So the question becomes what was "obvious" to Jesus / Matthew as the originator of the saying. This requires studying textbooks on New Testament background, which will also be useful for interpreting verses like walking the second mile, or the eye of the needle. What's not obviously literal/figurative to us may have been obvious to the original reader. If so, evangelical hermeneutics demands that THAT understanding should control the meaning of the text. In my short answer, I argued (though I admit, not with scholarly reference) that the original audience would have got it.

Secondarily, there is the "text-centered" meaning where the book-context of the text plays a role in providing the clue. For example, in John 6:55 case about eating Jesus's flesh as bread of life, there is a preceding context (feeding of the five thousand, manna), an internal play on words intended to shock the audience, a decidedly spiritual interpretation given within the chapter (words of eternal life, vv 60-65) that the author of John expects the reader to take into account when interpreting whether John 6:55 is literal.

Finally, in defense of coherency in spite of the seemingly inconsistent use of "obvious" and "literal", I recommend reading up theoreticians of meaning (such as Saussure and Derrida) so we become more sensitive on aspects of using language that we take for granted. For example, a good introduction to what's "obvious" in "literalism" from linguistics perspective, which applies our automatic use of "obvious" and "literal" in everyday language to various difficulties of Biblical literalism can be found in this short 2018 MA in theology thesis Explaining the Obvious: Privileged Hermeneutics and the Irony of Explicitly Literal Interpretation.

Applying "author-centered" meaning to Gen 1-11, the question becomes how the original redactor understood it? That's where Ancient Near Eastern research come into play, which lead us to interpret it mythically. Even when an NT author seemingly cite verses from Gen 1-11 using their literal meaning (Paul citing Adam in Romans 5, Jesus citing the sign of Jonah and Noah's flood), it does NOT automatically certify that Gen 1-11 cannot be interpreted mythically, because we have to ask what was precisely the intention of the NT author? Is there intertextual meaning at play here? Even if the NT author intended literal meaning, how about the element of divine accommodation that also influenced the redactor of Noah's flood story, for example?

Conclusion

There is no easy answer, but there is a coherence in Evangelical's privileging of the "literal" meaning as long as it is combined with good exegetical practices described above. In the history of Bible interpretation, "literal" here is not primarily contrasted with "mythical" / "figurative", but with an entirely different 3 other senses: allegorical / typological, moral / tropological and analogical. In this understanding of "literal", mythical meanings STILL lie within the "literal" sense, which is also privileged by most non-inerrantist Evangelicals although this is more flexible than the biblical literalism favored by Creationists.