This answer relates to the Church of Scotland.
Dancing has often been regarded with deep suspicion in the Presbyterian tradition. Originally, it was strongly associated with all kinds of bad behaviour, and would certainly not be permitted in church. Even after dancing became more socially acceptable, church was still meant to be a solemn and sober place. This is less true today, though there is still a lot of historical inertia against allowing dancing.
In 1649, the General Assembly of the Church of Scotland banned "promiscuous dancing":
The Assembly, finding the scandall and abuse that arises thorow promiscuous dancing; do, therefore, inhibit and discharge the same, and do referre the censure thereof to severall Presbyteries, earnestly recommending it to their care and diligence.
"Promiscuous dancing" particularly refers to mixed dancing at weddings and other celebrations, and was associated by the Kirk with drunkenness, gluttony, immodesty and lechery. Offenders could be punished by having to pay a fine, or be made to stand in the "place of repentance" at the front of the church, so that they could be rebuked during the sermon. Dancing on Sundays was particularly frowned upon, since Sunday was meant to be reserved for solemn religious observance. An earlier prohibition applied only to ministers, who were expected to set a moral example; the Second Book of Discipline (1578) said that Kirk sessions could excommunicate elders or ministers
that teach erroneous and corrupt doctrine; that are of scandalous life, and, after admonition, desist not; that are given to schism or rebellion against the kirk, manifest blasphemy, simony, corruption of bribes, falsehood, perjury, whoredom, theft, drunkenness, fighting worthy of punishment by the law, usury, dancing, infamy, and all others that deserve separation from the kirk. (7.16)
John Knox, in particular, disliked the feasting and dancing he saw at the court of Mary, Queen of Scots, telling her (in December 1562, according to his own account in Historie of the Reformation of Religioun within the Realm of Scotland) that "fiddling and flinging", instead of "reading and hearing God's most blessed word", was morally objectionable for a ruler. He went on to say that "the reward of dancers ... will be to drink in Hell, unless they speedily repent, so shall God turn their mirth into sudden sorrow".
If secular dancing was seen in such a light, it is not surprising that dancing in church was unimaginable. Instrumental music was often regarded in the same way, and there were disagreements over forms of singing: for example, whether polyphony or harmony were allowed, or whether religious music should stick to first-century forms instead. In time, it was recognized that the acceptability of music in (polite!) society made a difference to how it could be used in church - it was no longer automatically associated with debauchery and decadence. This did not quite extend as far as dancing: in fact, one of the counter-arguments to permitting more diverse forms of music was that it would lead to dancing in church as well.
One of the main reformers was Robert Lee, a minister at Greyfriars Kirk in Edinburgh. In 1860 he had an organ installed there - the first one at any Presbyterian church in Scotland. In his book The reform of the Church of Scotland in worship, government and doctrine (1864) he discusses Old Testament references to dancing, saying that it did not take place as part of Temple rituals, and therefore does not constitute a precedent for Christian churches:
We are only informed that the ancient Jews, in the excitement of their joy, actually danced on certain special occasions, such as after some great victory or some wonderful deliverance; but we never read that dancing was any regular or ordinary part of their religious worship. ... We have not a word to show that dancing enjoyed any divine sanction, or that any such thing was ever admitted or thought of as part of that ritual. Also, while "singers and players on instruments" are repeatedly spoken of in connection with the temple-service, no mention is ever made of dancers. ... The Seer of the Apocalypse also beholds "the elders" and "them that had gotten the victory," with "the harps of God" (chap. V and XV); but he says nothing of dancing; for no such thing was known in the temple-worship, which forms the basis of his imagery. It is in vain, therefore, to pretend that dancing holds the same position in the Old Testament ritual as the use of musical instruments.
He further notes that there is no New Testament reference to dancing in worship. Lee also proposes a general principle for how ancient modes of worship may be received and adapted, maintaining the essence but without having to adhere to the same exact form:
In all these cases, and in others which might be named, we not only consider what was practised by the ancient Jews, even under the sanction of Divine authority, but what may be decent and reverent in us, in our different climate, with our different institutions, customs, manners, and associations, and national character. ... These things belong not to the essence of religion, but are merely means towards an end, and are therefore to be judged of, in all cases and at all times, according to their fitness for promoting that end.
Indeed, John Calvin had written (Institutes 3.20):
If singing is tempered to a gravity befitting the presence of God and angels, it both gives dignity and grace to sacred actions, and has a very powerful tendency to stir up the mind to true zeal and ardor in prayer. We must, however, carefully beware, lest our ears be more intent on the music than our minds on the spiritual meaning of the words. ... Songs composed merely to tickle and delight the ear are unbecoming the majesty of the Church, and cannot but be most displeasing to God.
This would apply just as well to dancing as to singing. In general, contemporary arguments about dancing probably depend very much on the nature of the dancing and of its context in worship, as opposed to how well particular forms of dance (or singing, or instrumental music) mirror first-century practice. For example, we could argue that dance tends to be a performance for others to watch, rather than being a truly participatory act of worship. But the historical argument against it is not a knock-out blow, given that we have permitted many other innovations in worship since that time, even just in the domain of music.