Probably not specifically Irish, s it seems to have been quite well known in England too.

Great Wolford Mummers Play collected 1913, Warwickshire England
In comes I Fidler Wit
My head's so large, me wits so small
I've brought me fidler to please you all.
Toll-de-roll the tinder box
Father died the other night
And left me all his riches,
A wooden leg, a feather bed,
And a pair of leather breeches,
A coffee pot without a spout,
A jug without a handle,
A guinea pig without a wig,
And half a farthing candle.
Sing brothers sing.

Stony Play, Stony Stratford, Buckinghamshire England
(To the tune of 'The Red Flag')
Father died the other night and left me all his riches
A wooden leg, a feather bed, a pair of leather breeches
A coffee pot without a spout, a jug without a handle
A guinea pig without a wig, and half a farthing candle
Sing brothers sing

Not knowing which tune was originally used, the Shakespeare Mummers sang it to Yankee Doodle, but now use Brighton Camp instead

I believe there is a much longer song on a similar subject in the DT but I can't find it