Looking at the various sources, it could indeed be burglar's wine (doped wine supposedly used by dishonest innkeepers so they could more easily steal their patrons' valuables while they slept).

But since the song was originally collected in West Virginia (Folk Songs of the South, a collection of West Virginia songs edited by John Harrington Cox, prints a version collected in 1915, and says it was popular in the area in the 1890s), a persuasive argument is that burglar's wine, Burgundy wine, etc. are likely garbled forms of burgaloo wine, which was apparently (Dictionary of Americanisms, John Russell Bartlett, 4th ed., 1889) a type of pear wine, burgaloo being a variant spelling or mispronunciation of virgelieu, a popular pear variety of the 19th century in West Virginia. But the term burgaloo wine, having become obsolete, folk singers would substitute words that made more sense to them.

My information was gathered from various sources on the Internet, including from an article by Lyle Lofgren, listed as having been reprinted from Inside Bluegrass, May 2003.