As Gareth Rees pointed out, the Copyright Act of 1911 was the copyright law that was in force when T. S. Eliot published "The Waste Land". Below is how this act defines "copyright" (emphasis mine):
For the purposes of this Act, "copyright" means the sole right to produce or reproduce the work or any substantial part thereof in any material form whatsoever, to perform, or in the case of a lecture to deliver, the work or any substantial part thereof in public; if the work is unpublished, to publish the work or any substantial part thereof ; and shall include the sole right,—
(a) to produce, reproduce, perform, or publish any translation of the work;
(b) in the case of a dramatic work, to convert it into a novel or other non-dramatic work ;
(c) in the case of a novel or other non-dramatic work, or of an artistic work, to convert it into a dramatic work, by way of performance in public or otherwise;
(d) in the case of a literary, dramatic, or musical work, to make any record, perforated roll, cinematograph film, or other contrivance by means of which the work may be mechanically performed or delivered, and to authorise any such acts as aforesaid.
Section 3 also states (emphasis mine),
The term for which copyright shall subsist shall, except as otherwise expressly provided by this Act, be the life of the author and a period Of fifty years after his death: (...)
Eliot's quotes in "The Waste Land" reproduce at most a few lines from other works, i.e. no "substantial parts". In addition, copyright had already expired for many of the works he quotes, e.g. Bible books, Elizabethan and Jacobean authors such as Shakespeare, Kyd and Webster, and the works of Dante.
So it was not copyright legislation that required such notes, but the perception of plagiarism by readers may have played a role. Peter Ackroyd's biography of T. S. Eliot notes (on pages 177-178 of the German translation, published by Suhrkamp in 1988):
Die Ausgaben in Buchform unterschieden sich von den Zeitschriftenfassungen durch die Beifügung von Eliots Anmerkungen. Diese waren ursprünglich dazu gedacht, Pagiatsvorwürfe, wie sei seinen früheren Gedichten gemacht wurden, zu unterlaufen. Da aber das Gedicht noch immer nicht lang genug schien, um ein Buch zu ergeben, weitete er sei aus. Was daraus hervorging, nannte er dann eine "beachtliche Zurschaustellung gelehrter Hochstapelei".
The book editions [of "The Waste Land"] differed from the magazine editions by the addition of Eliot's notes. These were originally intended to avoid the accusations of plagiarism that had been levelled at his earlier poems. Since the poem still seemed too short to fill a book, Eliot expanded the notes. He later called the result a "remarkable exposition of bogus scholarship".
The comment about "bogus scholarship" comes from his lecture "The Frontiers in Criticism", which was later published in On Poetry and Poets. Below is a longer quote from that lecture:
I had at first intended only to put down all the references for my quotations, with a view to spiking the guns of critics of my earlier poems who had accused me of plagiarism. Then, when it came time to print The Waste Land as a little book—for the poem on its first appearance in The Dial and in The Criterion had no notes whatever—it was discovered that the poem was inconveniently short, so I set to work to expand the notes, in order to provide a few more pages of printed matter, with the result that they became the remarkable exposition of bogus scholarship that is still on view to-day. I have sometimes thought of getting rid of these notes; but now they can never be unstuck. They have had almost greater popularity than the poem itself—anyone who bought my book of poems, and found that the notes to The Waste Land were not in it, would demand his money back.
So avoiding accusations of plagiarism played a role, but not in a legal sense.