This was actually attributed to General US Grant, who was famously tone-deaf.
Lincoln, on the other hand, was self-depricating about his dancing ability.
As a young man he was approached by an attractive young lady at a dance, who came up to him and said “Mr Lincoln, I would like to dance with you in the worst way.”
When the dance was over Lincoln told her “You got your wish. I did dance in the worst way.”
But back to Danny Boy, this is my favourite performance of that song: