It's just fiction, of course, but it's made obvious that he/she is dying from syphilis because of the mention of mercury, salivation, lock hospitals, flash girls etc. in early texts.
The whole point of the song seems to be a warning against catching diseases from whores.
I'm talking about older texts of the "Unfortunate Lad" type, though, in the "St. James' Infirmary Blues" (which has little in common with the Unfortunate Lad) the cause of death isn't specified at all.

It's not even clear why the guy is dying as well in "Infirmary".
We just hear that his girl dies in a hospital and then he starts making requests for his own funeral.