I'm not sure if this is what you're looking for, but you may be interested in the choruses of Aristophanes' plays, particularly Peace (421 BC) and Lysistrata (411 BC), which would have been sung to the audience of the play and contain some anti-war sentiments (though arguably only for the Peloponnesian War, rather than war and violence in general).

edit: A link to a translation of the first chorus of Peace.

The oldest I know right off my head is the song "Schwabenkrieg" from the author of the German anthem, Hoffmann von Fallersleben. Composer is Ernst Richter, date of the text is 1836.

There are also several songs from the thirty years war some of which are still in use today. They are not anti-war-song by their intent, but instead focus on the suffering of the lower classes.

I don't know if it could be described as "anti war" in the general sense but there are folk songs throughout Europe describing the privations of going to war, the womenfolk left behind, and the ruin that soldiers bring with them when they pass.

"Johnny Has Gone for a Soldier" goes back at least as far as the Revolutionary War, and the Wikipedia article on Folk Music of England notes

From roughly the same period [mid-17th century], songs of protest at war, pointing out the costs to human lives, also begin to appear, like "The Maunding Souldier or The Fruits of Warre is Beggery", framed as a begging appeal from a crippled soldier of the Thirty Years War.

The source quoted in the article is de Sola Pinto and A. E. Rodway, The Common Muse: An Anthology of Popular British Ballad Poetry, XVth-XXth Century (Chatto & Windus, 1957), pp. 39-51, 145, 148-50, 159-60 and 250

One of the earliest verses I can think of that may express the futility of war belongs to the conclusion a variant of the Scottish ballad "The Battle of Harlaw":

On Monanday, at mornin,
The battle it began,
On Saturday, at gloamin,
Ye’d scarce kent wha had wan.

An sic a weary buryin
I'm sure ye never saw
As wis the Sunday after that,
On the muirs aneath Harlaw.

Gin ony body speer at you
For them ye took awa,
Ye may tell their wives and bairnies
They're sleepin at Harlaw.

The last line of the first verse given here translates to "You could scarcely tell who'd won;" the first line of the third verse translates to "If anyone asks you." (The rest of it, I think, can be parsed even if your Scots is a bit rusty.) The Battle of Harlaw was fought in 1411, but it's impossible to date the composition of the ballad (this particular text was documented in 1888) or know when these verses entered the associated tradition.

Broadly speaking, I think that most songs that critique war in the Anglo vernacular song tradition are rhetorically focused on tales of personal suffering rather than through philosophical objections in the vein of Phil Ochs's "I Ain't Marching Anymore." Some songs that proved particularly successful in oral circulation (and that I also think are quite poignant, especially the former) are "The Banks of the Nile" (NSFW) and "Bonny Light Horseman."

There are, of course, a few exceptions to the personal depredation rule. There's "My Son John" (aka "My Son Tim," "Mrs. McGrath," and others), which proved successful in oral circulation, in which the mother often levels criticisms against "all foreign wars" when her son comes home minus some of his limbs. Perhaps most dramatically, there is also an early/mid-nineteenth century English broadside entitled "Thirteen Pence A Day" that puts a dry spin on the morality of soldiering:

When you are a soldier, lad, if you do not limp,
You shall go recruiting lad, then you'll be a crimp.
Perhaps a yokel takes your money, when in drink he's fresh,
But you know it's quite respectable to kidnap human flesh.

Come and learn your exercise, run lads, run.
Soon you'll know the use of bayonet and gun,
Then you'll go abroad my lads, and there you'll soon be warm,
By shooting men you never knew, who never did you harm.

I couldn't say, however, if this one ever entered oral circulation.

Although probably not the kind of song that you're thinking of, the 7th century BCE Greek lyric poet Arkhilokhos (Archilochus) is famous for short lyric poems that reflect war-weariness. One example:

Ἀσπίδι μὲν Σαΐων τις ἀγάλλεται, ἥν παρὰ θάμνῳ

ἔντος ἀμώμητον κάλλιπον οὐκ ἐθέλων·

αὐτὸν δ' ἔκ μ' ἐσάωσα· τί μοι μέλει ἀσπὶς ἐκείνη;

Ἐρρέτω· ἐξαῦτις κτήσομαι οὐ κακίω.

One of the Saiôn in Thrace now delights in the shield I discarded

Unwillingly near a bush, for it was perfectly good,

But at least I got myself safely out. Why should I care for that shield?

Let it go. Some other time I'll find another no worse.

Classical musicians have been indirectly critiquing war since the Renaissance. An example is the keening final chorus of Carissimi 's Jeptha, Plorate filli Israel, from the 1600s or so.

Interestingly, Mozart's last solo vocal work before his death was an anti-war cantata pleading for brotherhood among all men, written for his Masonic lodge. It dates to 1790.

Source: have performed it and researched it.

Folks, this thread is not for you to share trivia about old anti-war songs. If you don't actually know the oldest anti-war song, I would ask that you not try and answer.