I honestly hadn't considered where the headings came from. I think I always looked at them as the rest of the headings in my bible and without precisely contemplating it had considered them human as well. But I looked through my commentaries and found some discussion of this. Also, I asked a friend, Dr. Byron Curtis who teaches a course on The Psalms at Geneva College, and he gave me a good answer as well.
He says, "These headings are mysterious. The ancient textual versions are not at all agreed as to what these headings say, or even for which psalms they appear. The Septuagint (200 BC), which ordinarily reproduces a strict Greek equivalent of the underlying Hebrew text, varies greatly on the headings of the Psalms, and has many more headings than the standard Hebrew text. That fact brings the antiquity of the Hebrew psalm headings into doubt. The Hebrew headings seem to have solidified only after the Septuagint was made.I also found in Calvin's commentary a footnote included which said, "These titles are destitute of authority, as the careful reader of The Psalms will soon remark; they are to be regarded merely as marginal glosses of the Jews, but poor guides to the interpretation of Scripture." And credits "Fry's Translation and Exposition of the Psalms" with the quote.
I'm not at all convinced that the headings were an original part of the inspired text, and am inclined to support the widespread view among scholars that these headings are rather like the editor's headings that introduce the paragraph topics in versions like the NIV. When they were added to the Hebrew text, and by whom, remains unknown.
The other question is whether the headings accurately preserve the circumstances they allegedly report, or whether they represent interpretive "guesses" based on their understanding of the meaning of the text.
In other words, does the heading of Psalm 3, ascribing it to the time of Absalom's revolt, represent a bit of ancient exegesis by some ancient scholar of the psalms, or is in an authentic report of the circumstances in which the psalm was written? No one really knows the answer to that question. Sometimes the headings' contents seem very appropriate; for other psalms less so.
I would be very reluctant to describe the headings as part of the inerrant, inspired text."
Spurgeon includes this quote in his commentary on The Psalms, "With regard to the authority of the titles, it becomes us to speak with diffidence, considering the very opposite opinions which have been offered upon this subject by scholars of equal excellence. In the present day, it is too much the custom to slight or omit them altogether, as though added, nobody knows when or by whom, and is, in many instances, inconsistent with the subject matter of the Psalm itself: while Augustine, Theodoret, and various other early writers of the Christian church, regard them as a part of the inspired text; and the Jews still continue to make them a part of their chant, and the rabbins to comment upon them.
It is certainly unknown who invented or placed them where they are: but it is unquestionable that they have been so placed from time immemorial; they occur in the Septuagint, which contains also in a few instances titles to Psalms that are without any in Hebrew; and they have been copied after the Septuagint by Jerome. So far as the present writer has been able to penetrate the obscurity that occasionally hangs over them, they are a direct and most valuable key to the general history or subject of the Psalms to which they are prefixed; and, excepting where they have been evidently misunderstood or misinterpreted, he has never met with a single instance in which the drift of the title and its respective Psalm do not exactly coincide. Many of them were, doubtless, composed by Ezra at the time of editing his own collection, at which period some critics suppose the whole to have been written; but the rest appear rather to be coeval, or nearly so, with the respective psalms themselves, and to have been written about the period of their production. - John Mason Good, M.D. 1854
I think that sums it up pretty well and hopefully answered the question satisfactorily. They do not seem to be inspired in any way, although the vast majority of them have been used consistently from antiquity until now.
(Note: if you would like to look into this in your own commentaries as well please learn from my experience. I read many of the introductions to the Psalms and found not one mention of this issue. But in general if you will go to Psalm 3 you will find it there discussed. Save yourself some reading!)