There are a couple compelling cases for consideration in the Psalms of the Bible. Naturally, these come with all the conventional caveats regarding the relative historicity of these works (i.e., that they are of unknown authorship or are pseudepigraphically attributed and that their era of authorship can be given collectively as no better than belonging to a range of several centuries).
However, the great majority of these works strictly satisfy the characterisation "song" (with considerations relevant to musical performance directly addressed therein), and certain among them address political considerations which could be deemed messages of protest.
Psalm 137, eminently familiar to most via its opening passage
By the rivers of Babylon, there we sat down, yea, we wept, when we remembered Zion
is an interesting case, as not only is it, itself, a declaration of political grievances (albeit addressed from an after-the-fact standpoint), but moreover it specifically and internally addresses the difficulty of singing songs of praise, when one would more so be driven to sing songs of protest, and thereupon gives an example of such. To wit,
O daughter of Babylon, who art to be destroyed; happy shall he be, that rewardeth thee as thou hast served us.
Happy shall he be, that taketh and dasheth thy little ones against the stones.
(Pardon my fondness for the KJV, incidentally).
While without credible authorial attribution or provable date of authorship (but 5th or late 6th century BCE being the implication), such a psalm's implicit claim to authorship in the early Postexilic Period is itself perfectly credible. The era in which this psalm thematically places itself is the period in which we would otherwise be inclined to assume it was composed. There is no anachronism here.
It feels likely that there are older examples, but as a start: the Diggers movement in the mid 17th century produced a number of songs we can unambiguously see as protest music. For those unaware of the Diggers, they were an agrarian socialist collectivist movement that sprang up during the chaotic times around the English civil war (I've heard them described as the forerunners of modern anarchism).
A good example is The Diggers Song, which talks about bringing down the gentry/priests/etc (if anyone has any other protest songs of this era, please PM me as I'd love to learn then). The same story is perhaps more familiar in the C20th folk-revival song World Turned Upside Down.
I'm not really aware of any protest songs as explicit as this from earlier periods. Some have claimed that The Cutty Wren dates back to the Peasants' Revolt, but the evidence for this is scant. Even if correct, it's a far more metaphorical text than protest songs usually are.
(If this is too speculative, my apologies: I'm a singer who incidentally studies history rather than a historian. I can add some citations when I get home, but most of those books are also written by musicians rather than historians, so it's hard to tell where historical record ends and personal theories begin.)