The answers by Josh61 and FumbleFingers provide a solid baseline notion of when “state of the art” arose in three senses: “status of the art” (late nineteenth century, according to WorldWideWords, citing the OED, in Josh61’s answer); “current stage of development of a practical or technological subject” (1910, according to Wikipedia, also citing the OED, in Josh61’s answer); and as a hyphenated adjective phrase meaning something like “top-of-the-line” or "best available" (1955, according to FumbleFingers’s answer).
A hike through Google Books search results turns up earlier instances of the phrase in each sense.
‘State of the art’ as ‘status of the art’
First, as noted in the body of the poster’s question, the “status of the art” sense of the phrase occurs in Charles Burney, A General History of Music, From the Earliest Ages to the Present Period, volume 4 (1776), referring to the French writer and composer Jean-Benjamin de La Borde as
a writer who, on some occasions, seems to know better, and to have ideas of good Music, more worthy of a master of harmony and the present state of the art in every part of Europe.
(The hostility of de La Borde toward Jean-Jacques Rousseau’s music criticism, may have been more political than objectively theoretical: de La Borde was a court musician and composer, and perished in 1794 during the Reign of Terror in the aftermath of the French Revolution.)
Earlier still is this instance from Lecture III in Thomas Sheridan, Lectures on the Art of Reading; First Part: Containing The Art of Reading Prose (1775):
There is not any thing which can shew the low state of the Art of Reading among us, in a stronger light, than the general complaint, that the service of the church is so seldom delivered with propriety. At first view, one would be apt to imagine, that in the settled service, open to all to be studied and examined at leisure, every one, by suitable pains, might make himself master of the proper means of reading it.
Thirteen years earlier, in A Course of Lectures on Elocution (1762), Sheridan had used the phrase “state of the art” in connection with writing:
On this account [namely, the tendency of readers to speak more monotonously than illiterate people] it is, that the most bookish men are generally remarkable for the worst delivery : as reading therefore by means of the press, is become almost universal amongst us ; and as the chief errours and defects of our delivery, arise from a faulty manner of reading ; there can not be a matter of more importance, than to explain fully, how this faulty manner, must necessarily prevail, not only from the unskilfulness of masters, but also from the imperfect state of the art of writing itself, until a proper remedy be found.
Also very early is this item from The Annual Register, or a View of the History, Politicks, and Literature, for the Year 1763, second edition (1765):
Their [Poland’s] military force consists, chiefly, in the Pospolite, that is, the whole body of the gentry, which, upon extraordinary occasions, the king and the national general can order into the field to serve for a limited time. The inconvenience and inutility of this military institution, in the present state of the art of war, need not be insisted on.
These instances indicate that the phrase “state of the art” meaning “status of the art” was already in use by 1765. The three earliest matches in a Google Books search are from England, and the arts involved are writing, war, reading, and music.
‘State of the art’ as ‘stage of technological development’
A determined advocate could insist that the reference to “the present state of the art of war” in The Annual Register for 1763 refers not to the status of war as an art but to the stage of technological (or at least strategic and logistical) development of war. Nevertheless, the sense in that example seems far more on the inefficacy of the system under which the Polish military operated than on the technological level of its armaments and equipment.
But even under a very narrow construction of the “state of the art” in the sense of “current stage of development of a practical or technological subject,” it is difficult to exclude this example from William Robertson, The History of the Discovery and Settlement of America (1777):
These voyages [by Phoenicians, Carthaginians, and Eudoxus of Cyzicus], if performed in the manner which I have related, may justly be reckoned the greatest effort of navigation in the ancient world ; and if we attend to the imperfect state of the art at that time, it is difficult to determine whether we most admire the courage and sagacity with which the design was formed, or the conduct and good fortune with which it was executed.
Likewise, from a review of R.E. Raspe’s translation of Inigo Born, Travels Through the Bannat of Temeswar, Transylvania, and Hungary in the Year 1770 in The Critical Review (April 1777):
The preface to that work [Ferber’s Letters] was a general view of the late improvements in mineralogy, and a plan for its future enlargement. In the same manner, in the preface to Born's Letters, Mr. Raspe gives a prospect of the present state of the art of mining, points out the means of bringing it to perfection, and describes the metallic mountains and their veins.
Another work by R.E. Raspe, “Account of the Present State and Arrangement of Mr. James Tassie’s Collection of Pastes and Impressions from Ancient and Modern Gems” (1786) speaks of engravings made on “hard and precious stones” under technologically primitive conditions:
They [“the lately discovered South Sea islanders”] will, like our barbarous ancestors in Europe, generally set a high value upon these first essays of art [namely, “stone weapons, tools, and ornaments”], and for good reasons. Are they not actually the most precious and costly things they are possessed of? Their safety, their preservation, and most of their useful arts, depend on them ; and the trouble of making a single hatchet must be very tedious and great indeed ; it must engage them continually for months and years. In that self-taught infant state of the art of engraving, they have nothing but the active powers of their hands, with sand and other hard stones, to conquer the almost invincible hardness of the materials.
Published the same year was Francis Grose, Military Antiquities Respecting a History of the English Army, from the Conquest to the Present Time, volume 1 (1786):
The military surgeons of ancient times are very little mentioned in history, perhaps they were not in very great estimation, the superstitious abhorrence of what was deemed a violation of the dead, prevented their having an accurate knowledge of the human frame, which is only to be acquired by frequent dissections : the practice of those times seems to have been confined to the composition of certain oils, balms and balsams, prepared with the grossest superstition, and administered under the rules of astrology. The low state of the art of surgery in France, even so late as the time of Francis I. contemporary with Henry VIII. may be gathered from the following note. [There follows a rather horrifying account of the practice of cauterizing gunshot wounds, written some time after a military campaign of 1536.]
And from the 1787 journal of Captain Cooper] of the East India ship Atlas cited in J Rennell, Observations on a Current that often prevails to the Westward of Scilly; Endangering the Safety of Ships that approach the British Channel,” in Philosophical Transactions, volume 83 (1793):
It would be worth perhaps the attention of government to send a vessel with time-keepers on board, in order to examine and note the soundings between the parallels of Scilly and Ushant at least ; from the meridian of the Lizard point, as far west as the moderate depths extend ; I mean such as can be ascertained with exactness in the ordinary method of sounding. I have reason to suppose that our chart of soundings is very bad; and indeed how can it be otherwise, considering the imperfect state of the art of marine surveying at the time when it was made?
Another early Google Books match is from Alexander McDonald, A Complete Dictionary of Practical Gardening, volume 1 (1807):
While almost every other department of useful science has been arranged and brought into a more accessible and convenient form, in the shape of a Dictionary, that of Gardening has remained nearly without assistance in that respect. The present is, however, an attempt to render so important and beneficial a branch of knowledge more easy and comprehensible, ... The execution of this undertaking, from the difficult and imperfect state of the art, has been attended with considerable labour and trouble; but the author hopes, from the practical knowledge which he possesses, and the various sources of information of which he has been enabled to avail himself, that it has been performed in a manner that will not be found less useful in directing the practical Gardener, than those who are not so conversant with the nature of the subject, ...
And from Theophilus Jones, A History of the County of Brecknock, volume 2 (1809), referring to the methods of coal and iron mining in use in the vicinity of Brecknock in ancient times:
... here it is worth remarking that as the charcoal was much the most unwieldy as well as perishable requisite for the operation, the ore was certainly carried to the charcoal rather than the fuel to the ore : these spots therefore, many of which are now, not within a mile of a bush, were at that time contiguous to, if not covered with wood ; beyond this there is little in these reliques worth notice, except as shewing the imperfect state of the art by which so large a portion of metal was left in the scoria, and the very low state or rather total want of commercial intercourse in those days, for when a farmer found that to quit his daily employment and turn metallurgist was an easy, it must certainly have been the only mode of procuring the iron he wanted.
In more civilized or less turbulent times, when internal communications were more safe and a degree of mercantile confidence was established, and when iron as well as other articles could be sold, it then became worth while to individuals to set up the manufacture as a distinct trade and to become iron masters. Observation and experience suggested, both in the process and in the machinery for carrying it on, improvements totally incompatible with the old ambulatory state of the art and the attention of the workmen in larger and more permanent establishments considerably abridged manual labour and rendered the process more frugal as well as more complete.
The earliest discussion of a “state of the art” for a particular science in Google Books search results is from 1777, and the sciences (or technologies) discussed in the earliest matches include navigation, mining, engraving, surgery, marine surveying, gardening, and metallurgy.
‘State-of-the-art’ as an adjective phrase
The earliest instance of “state-of-the-art” as an adjective phrase that a Google Books search finds is in George Woodling, Inventions and Their Protection (1938) [combined snippets]:
54. State-of-the-art search.—A state-of-the-art search is broader in scope than the preliminary search, and ordinarily is required to assist the engineering department in planning a new line of products. It usually is made where the engineering department has not as yet developed any definite ideas regarding the design of the new products. The purpose of the state-of-the-art search is to give the engineering department a general idea of the patent situation regarding the proposed new products before they go ahead with their detailed designs. With such assistance, the engineering department may make a comprehensive study of all the prior patents which may have a direct bearing upon the final design, and begin developments where the prior art left off. In this way there is no overlapping or duplication between the new development and prior inventions. When making a state-of-the-art search, the searcher orders patents freely and searches several related subclasses in order to give the engineering department a wide and general knowledge in the field.
A state-of-the-art search should not be confused with an infringement search. In a state-of-the-art search there is no effort made to determine whether the new products, yet to be designed, will be free of infringement when introduced on the market. The main purpose is to help the designer in working out a satisfactory line of new products.
(The 1938 edition of this book is also available, in full view, at Hathi Trust.)
The use of "state of the art" in the sense of "status of the art" or "stage of development of the science or technology" is considerably older than some sources have suggested, dating back to the 1700s in both instances. Likewise, "state-of-the-art" as a compound adjective is somewhat older than one might have expected, with a first occurrence in Google Books search results of 1938. Nevertheless, use of the phrase in the particular (and now very common) sense of "the best currently attainable (or obtainable)" does not appear to be appreciably (if at all) older than the dates given by the sources that Josh61 and FumbleFingers cite.