The place to start is always with the notes to the three-volume Kinsley Poems and Songs (1969), for poetic manuscripts, and with the notes for each letter in the two-volume Roy Letters (1985) for the manuscripts of letters. More up to date than Kinsley for poetic manuscripts, though still twenty years old, is:
Margaret M. Smith, ed., Index to English Literary Manuscripts, vol. 3, 1700-1800, part 1 (London; Mansell, 1986): Rare Reference Z 6611.L7 I5 [shelved in Smith Reading Room]
The sites linked in the panel above also document the locations for many individual Burns manuscripts.
For some poems, multiple manuscripts survive, with fair copies made by Burns for friends or patrons. For others, there may be just one manuscript, or none, with the only source for the text being a transcript (copy) by a later editor or collector, or an early printed version based on a manuscript that is now lost. Note that Burns's handwriting changed over time, and it is important to look at manuscripts from the same or an adjacent year in finding material for comparison.
Although most researchers now look first for digital facsimiles, some of the earlier facsimiles in book form, listed below, remain important. For a discussion, see the link "Some Burns Manuscripts in Facsimile," in the column on the left.
J. C. Ewing and Davidson Cook, eds., Robert Burns's Commonplace Book (orig. 1938; repr. Carbondale: Southern Illinois UP, 1965): PR4307 .C6 1965
Desmond Donaldson, ed., The Glenriddell Manuscripts of Robert Burns (Hamden, CT: Archon, 1973): PR4300 1973 .H3
The Geddes Burns (Boston: Bibliophile Society, 1908) [a facsimile of the copy of the 1787 Poems, with 27 pages of additional manuscript poems written by Burns for Alexander Geddes]: PR 4300 1787a .E4
[W.L. Bixby et al.], Poems and Letters in the Handwriting of Robert Burns reproduced in facsimile (St. Louis: for the Burns Club, 1908): PR 4302 .S6
[R.B. Adam], Autograph Poems and Letters of Robert Burns (Buffalo: privately printed, 1922; printed transcripts, rather than facsimiles): PR4302 .A3 1922
Peter Westwood, ed., The Definitive Illustrated Companion to Robert Burns, 8 vols. ([Irvine]: [privately published in connection with the Distributed National Burns Collection Project], 2004): PR4300 2004 .W4
“The Edinburgh Forgeries”, Burns Chronicle, 1st ser., 2 (1893): 123-134.
“The Sequel to the Edinburgh Forgeries,” Burns Chronicle, 1st ser., 3 (1894): 139-140.
William Roughead, “”Antique Smith," in his The Riddle of the Ruthvens and Other Studies (orig. 1919: revd, ed. Edinburgh: Moray Press, 1936), 122- 143.
Henry T. Scott, “Ch. XXXII: Wholesale Forgeries Perpetrated in Edinburgh,” in his Autograph Collecting: A Practical Manual for Amateurs and Historical Students (London: L. Upcott Fill, 1894), c171-180.
J. DeLancey Ferguson, “’Antique’ Smith and his Forgeries of Robert Burns,” The Colophon, A Book Collector’s Quarterly, 4, no. 13 (February 1933), unpaginated.
Gerard Carruthers and G. Smith,“Daylight Rabbery: The story of Antique Smith's Robert Burns forgeries,” The Drouth, 44 (2013): 10-15.