Those who argue against images of Jesus do so primarily in two ways: (1) that any image of Christ is necessarily inadequate and false and (2) that images of Christ inspire worship and devalue the Word of God.

Images of Christ inadequate and false

Advocates of this position regularly appeal to the incomprehensibility of God. John Calvin¹ and J. I. Packer² both refer to Isaiah 40, particularly verse 18:

To whom then will you liken God, or what likeness compare with him? (ESV)

Purported images of Jesus, therefore, are necessarily inadequate portrayals. Specifically regarding images of Jesus on the cross, Packer writes:

[The crucifix] displays His human weakness, but it conceals His divine strength; it depicts the reality of His pain, but keeps out of our sight the reality of His joy and His power. In both these cases, the symbol is unworthy most of all because of what it fails to display.²

John Murray similarly argues that we have been given no information about the appearance of Jesus, and therefore have no right to make an image of him:

We have no data whatsoever on the basis of which to make a pictorial representation; we have no descriptions of his physical features which would enable even the most accomplished artist to make an approximate portrait. [...] No impression we have of Jesus should be created without the proper revelatory data, and every impression, every thought, should evoke worship. Hence, since we possess no revelatory data for a picture or portrait in the proper sense of the term, we are precluded from making one or using any that have been made.³

But, some argue, Jesus had a human body, so couldn't an image of that be made? James Fisher and Ebenezer Erskine respond that pictures of Christ are "downright lies, representing no more than the picture of a mere man":

Though he has a true body and a reasonable soul, yet his human nature subsists in his divine person, which no picture can represent. [...] The true Christ is God-man.⁴

Images of Christ inspire worship and devalue the Word of God

Another concern expressed by opponents of images of Christ is that any image of Christ, if meaningful at all, necessarily inspires worship. John Murray writes:

A picture of Christ, if it serves any useful purpose, must evoke some thought or feeling respecting him and, in view of what he is, this thought or feeling will be worshipful. We cannot avoid making the picture a medium of worship. [...] It is a grievous sin to have worship constrained by a human figment, and that is what a picture of the Saviour involves.³

Thomas Vincent's language is often quoted:

If it do not stir up devotion, it is in vain; if it do stir up devotion, it is a worshipping by an image or picture, and so a palpable breach of the second commandment.⁵

Rather than using images, proponents argue, God reveals himself to us through his Word, and to that we ought to limit ourselves. Explaining Deuteronomy 4, part of which is also cited by the Westminster Assembly in support of this position, Packer writes:

God did not show [the people of Israel] a visible symbol of Himself, but spoke to them; therefore they are not now to seek visible symbols of God, but simply to obey His word. [...] To make an image of God is to take one's thoughts of Him from a human source, rather than from God Himself; and this is precisely what is wrong with image-making.²

Fisher and Erskine, following the Westminster Assembly, cite Romans 1:21–23 as referring to those who create images of either God or Christ, even in their own minds:⁴

they became futile in their thinking, and their foolish hearts were darkened. [...] [they] exchanged the glory of the immortal God for images resembling mortal man and birds and animals and creeping things. (ESV)

Summary

Murray restates his concerns with images of Jesus as follows:

In summary, what is at stake in this question is the unique place which Jesus Christ as the God-man occupies in our faith and worship and the unique place which the Scripture occupies as the only revelation, the only medium of communication, respecting him whom we worship as Lord and Saviour. The incarnate Word and the written Word are correlative. We dare not use other media of impression or of sentiment but those of his institution and prescription. Every thought and impression of him should evoke worship. We worship him with the Father and the Holy Spirit, one God. To use a likeness of Christ as an aid to worship is forbidden by the second commandment as much in his case as in that of the Father and Spirit.³

Images of Jesus, thus, fail to account for his divine nature, downplay the revelation of Scripture, and encourage worship not in accordance with God's law.

References:

1. Institutes, 1.11.2
2. Knowing God (1973), Chapter 4
3. "Pictures of Christ," from Reformed Herald, February 1961.
4. The Westminster Assembly's Shorter Catechism Explained, Question 51
5. An Explicatory Catechism, Question 51