See the Folk Music Journal article I mentioned a few posts (and nearly two years) ago. Burns slightly re-wrote an existing text (presumably from tradition), but didn't publish a tune; so his contribution would be insignificant if not for his fame.

Several distinct tunes have been found with forms of the song in "modern" times (over the last 150 years, say), the best known of which -and likely the one you mean- is probably the ubiquitous and protean Dives and Lazarus.