The phrase to bury (or to hide, etc.) one’s head in the sand means to refuse to face up to unpleasant or awkward realities.

This phrase refers to the practice traditionally attributed to the ostrich of thrusting its head into the sand when being overtaken by pursuers, supposedly through an incapacity to distinguish between seeing and being seen.

Other practices attributed to the ostrich include:
– an indiscriminate voracity, in particular a liking for hard objects which the ostrich swallows to assist the action of the gizzard;
– a lack of regard for its young, the eggs being partly hatched by the heat of the sun, which has led to the belief that the ostrich deserts its nest.

In Naturalis Historia (The Natural History – 77), a vast encyclopaedia of the natural and human worlds, the Roman statesman and scholar Pliny the Elder (23-79) wrote that the ostrich displays stupidity in hiding its head:

The history of the birds follows next, the very largest of which, and indeed almost approaching to the nature of quadrupeds, is the ostrich of Africa or Æthiopia. This bird exceeds in height a man sitting on horseback, and can surpass him in swiftness, as wings have been given to aid it in running; in other respects ostriches cannot be considered as birds, and do not raise themselves from the earth. They have cloven talons, very similar to the hoof of the stag; with these they fight, and they also employ them in seizing stones for the purpose of throwing at those who pursue them. They have the marvellous property of being able to digest every substance without distinction, but their stupidity is no less remarkable; for although the rest of their body is so large, they imagine, when they have thrust their head and neck into a bush, that the whole of the body is concealed.
—From The Natural History (London: Taylor and Francis, 1855), translated by John Bostock and Henry T. Riley.

However, according to the Greek historian Diodorus Siculus (fl. 1^st century BC) in Bibliotheca historica (Historical Library), far from displaying stupidity in hiding its head, the ostrich adopts a wise precaution:

The struthocameli [note 1] […], as their name implies, embrace in their form the compound of a bird​ and of a camel. For in size they are like a newly-born camel, but their heads bristle with fine hair, and their eyes are large and black, indistinguishable in general appearance and colour from those of the camel. It is also long-necked and has a beak which is very short and contracted to a sharp point. And since it has wings with feathers which are covered with a fine hair, and is supported upon two legs and on feet with cloven hoofs, it has the appearance of a land animal as well as of a bird. But being unable by reason of its weight to raise itself in the air and to fly, it swiftly skims over the land, and when pursued by hunters on horseback with its feet it hurls stones as from a sling upon its pursuers, and with such force that they often receive severe wounds. And whenever it is overtaken and surrounded, it hides its head in a bush or some such shelter, not, as some men suppose, because of its folly and stupidity of spirit, as if it thought that since it could not see the others it could not itself be seen by others either, but because its head is the weakest part of its body it seeks a shelter for it in order to save its life; for Nature is an excellent instructor of all animals for the preservation not only of their own lives but also of their offspring, since by planting in them an innate love of life she leads successive generations into an eternal cycle of continued existence.
—From Library of History (Cambridge, Massachusetts: Harvard University Press, 1935), translated by C. H. Oldfather.

The phrase to bury (or to hide, etc.) one’s head in the sand first occurs in simile form with explicit reference to the ostrich—the earliest instances that I have found are as follows, in chronological order:

1-: From Something Occasional Fatall Accident Blacke Friers (1623)—as quoted in the Oxford English Dictionary (online edition, December 2021), which, unfortunately, does not provide more context:

Like the Austridge, who hiding her little head, supposeth her great body obscured.

2-: From Mallet’s “William and Margaret” vindicated, published in The Gentleman’s Magazine: And Historical Chronicle (London, England) of March 1792—a correspondent signing themself ‘A. S.’ reacted to the comments that the English author and translator ‘Hesiod’ Cooke (Thomas Cooke – 1703-1756) had once made on William and Margaret, a poem by the Scottish poet and playwright David Mallet (1701/2-1765):

This malignant commentator then passes over in silence the solemn, the startling invocation; the touching questions and reproaches of the injured spirit; so transcendently natural, and of such heart-affecting simplicity, that no future plaint from a love-stricken maid, or from her upbraiding apparition, can rival them in interesting pathos—Yes, he passes over them in the silence of conscious envy, hoping, perhaps, that because he mentions them not, their excellence will be invisible, like the ostrich, which, when pursued by its hunters, thrusts its head into the sand, and fancies that, because it will not see them, they cannot see it.

3-: From Friendly Advice from a Minister to the Servants of his Parish (London: Sold by J. Deighton, 1793), by the Anglican priest Richard Cecil (1748-1810):

Such as think to avoid a danger by turning their eyes from it, have been well compared to that silly bird, who, when closely pursued, thrusts his head into the sand or a thicket: and because he does not see his pursuers, vainly hopes that they have lost sight of him.

4-: From the Morning Advertiser (London, England) of Tuesday 25^th May 1819:

WESTMINSTER ELECTION.

Yesterday nearly 300 Electors of Westminster celebrated the Twelfth Anniversary of what is termed Purity of Election, at the Crown and Anchor Tavern. […]
[…]
Mr. Hobhouse returned thanks in a long and able speech. […] A certain House was so old a strumpet, and so hackneyed in vice, that she could not recollect that ever she was a maid. She had her bullies about her, who fed upon the profits of her prostitution, and were ready to knock down any one who said a word against her purity; but notwithstanding their heroism, she was as well known as if the Society for the Suppression of Vice hung their lamp at her door.—(Loud laughter and applauses.)—Like the Ostrich, who thought that by hiding her head in the sand, she was unseen by all, because unseen by herself, she concealed all but those parts which should not be seen, and, to use the language of Edmund Burke, exhibited nothing to the people but the shameful parts of the Constitution [note 2].

A different transcript of Hobhouse’s speech appeared in The Statesman (London, England) of Wednesday 26^th May 1819:

It was plain enough that no Parliament was ever so set against Reform as the present. When any of them got to propose a motion, he was instantly callenged [sic] with the crime of intending it for Reform; to which the instant reply was, “No, I give you my honour it has nothing to do with Reform.”—(Laughter.)—Begging pardon of the Ladies (there were a few in the gallery), corruption was the oldest strumpet in Christendom; it might be said to be in the condition of Petronius’s heroine [note 3], who had been debauched so long she could not remember when she was a maid.—(Loud laughter.)—Yet such was the audacious arrogance of this venerable harlot, that she would not suffer a soul to breathe a hint of her impurity.—There were several prudes in the house (he could not help mentioning Mr. Wynn), who, in spite of her flagitious lewdness, would have them treat her as a being of untainted chastity, who would not hear a single remark in disparagement of her purity, though the contrary was as well known as if the Society for the Suppression of Vice had hung up a lantern at the door.—(Much laughter.)—This was like the cunning of the ostrich, who thrusts his head into the sand and leaves her tail wholly exposed, insomuch that, to borrow a phrase from Burke, they had become quite familiar with the shameful parts of the Constitution.—(Applause.)

Notes:

1 The Latin noun strūthĭŏcămēlus denotes the ostrich. This noun is from Greek στρουθοκάμηλος (strouthokámēlos), from στρουθός (strouthós), sparrow, and κάμηλος (kámēlos), camel.

2 This refers to the following from Speech on American Taxation, 1774, by the British man of letters and politician Edmund Burke (1729-1797):

A noble lord, who spoke some time ago, […] has said that the Americans are our children, and how can they revolt against their parent? He says, that if they are not free in their present state, England is not free; because Manchester, and other considerable places, are not represented. So then, because some towns in England are not represented, America is to have no representative at all. They are ‘our children’; but when children ask for bread we are not to give a stone. Is it because the natural resistance of things, and the various mutations of time, hinders our government, or any scheme of government, from being any more than a sort of approximation to the right, is it therefore that the colonies are to recede from it infinitely? When this child of ours wishes to assimilate to its parent, and to reflect with a true filial resemblance the beauteous countenance of British liberty; are we to turn to them the shameful parts of our constitution? are we to give them our weakness for their strength? our opprobrium for their glory; and the slough of slavery, which we are not able to work off, to serve them for their freedom?
—From The Works of the Right Honourable Edmund Burke (Oxford University Press, 1906).

3 This refers to Quartilla, a Priapic priestess in the Satyricon, attributed to Gaius Petronius (died AD 66).