I've been struggling to find any more lyrics for it, but here's a good start on the origins:

To emphasize the careless, irresponsible mood of the Davidites and their contempt for the conventions, traditions and critical standards of the Philistines, Schumann has woven into the march [March of the Hosts of David Against the Philistines] very cleverly a quaint old tune of the 17th century, known throughout Germany as the Grossvatertanz (Grandfather-dance), and a favorite college song at the German universities. It was also adopted in this country [the United States] and is familiar to those whose memories reach back over half a century, sung to the following dofferel:

Tim Doolan he dreamt that his father was dead,
And his father he dreamt that Tim Doolan was dead,
 And Tim Doolan was dead
 And his father was dead
And Tim Doolan he dreamt that his father was dead. (via Stories of Standard Teaching Pieces, by Edward Baxter Perry, pp. 61-62)

This song is tricky to track down because of the frequent variation used in repeating it, especially among the titular characters.

A May 17th, 1834 copy of The Mercury (Charleston, SC) contained the following quip in response to a distasteful editorial:

I have it--the Editor must be a descendent from an ancient family of dreaming noteriety, a remarkable instance of which is recorded in the following graceful and polished lines by the family poet.

John Dowling he dreamt that his daddy was dead,
And his daddy he dreamt that John Dowling was dead.

Had the poet survived to the present day, he might have added, and

Dan Dowling he dreamt that the College was dead. (link)

Aside from passage not worth quoting which appeared in the February 2nd, 1894 The Kirwin Globe (p. 4), the September 7th, 1899 The Kansas Patron (p. 1), and The Stockton (KS) Review on November 21st, 1912 (p. 6), which exchanged the name Tim Doolan (or John Dowling) for Dad Desbrow, John Dobbin, and John Dadden, respectively, there's only one other mentioned of this family of songs I could find.

In response to the suggestion that the city organist replace classical music with pop tunes, a poem was written which included the couplet "If Mozart were cut, and they heard in his stead / 'Tim Doolan didn't know that his father was dead.' " (The (Sydney, NSW) Bulletin, Vol. 35 No. 1787 (May 14, 1914), p. 36.

So in general, there doesn't seem to be much more going for it lyrically that I could find, but at least there's a start to an answer and a few examples of it historically.