I think “mud” may just refer to the wet, boggy, peat soil of Ireland.
And the place where all the rivers run dry is probably Hell.
The song strikes me as a defiant statement of Irish Catholic nationalism and contempt for Protestant England.
The lyrics aren’t clear or straightforward, but I think the basic meaning is, “If I ever stop fighting for Ireland, if I ever give in and accept rule by those foreign, Protestant bastards… well then, I’ve lost my soul. So, bury me in the peat and let me sink all the way to Hell.”